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U. S. Naval Postgraduate School

Monterey, California

THE LIFT COEFFICIMT OF FLAT PLANING SURFACES

A Thesis Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Science

U. S. Naval Postgraduate Schqol
STE7MS INSTITUTE OF TECHNOLOGY Njonterey. Cnlifornia

Submitted by Donald David Farshing, Jr.

TABLE OF CONTENTS

Page

Abstract

Notation

Introduction 1

Modification of the Two-Dim ens ional Planing Solution for

the Effect of a Finite Span in a Wfeightless Fluid 3

Historical Background 3

The Two -Dimensional Solution 3

Application of the Momentum Theorem - Method I 6

Application of the Momentum Theorem - Method II III

Experimental Investigation 16

Purpose 16

Equipment and Procedure 16

EjqDerimental Results 21

Comparison of Theoretical Results with Experimental Data 22

General Discussion 22

Empirical Correction to the Curves of C, Computed by

the Cubic Equation 22

The Effect of Gravity 2ii

Summary and Conclusions 2?

Bibliography 31

Appendix I 3^

Appendix II h^

Tables ^k

Figures

^S'7'5'^

ABSTRACT

A theoretical and experimental investigation bo determine the
lift coefficient of a flat plate planing surface covering a broad
range of trim angles and aspect ratios is presented.

The analysis is divided into two principal parts: first con-
sidered is the modification of the two-dimensional potential flow
solution for the effect of a finite span neglecting gravity; then,
second, the effects of gravity on the finite span planing surface
are discussed with the importance of Froude Number emphasized.

The analysis made on the effect of a finite span neglecting
gravity is accompanied by an experimental investigation in which
angles of trim up to 30 degrees and wetted length -beam ratios up to
about seven are explored.

The analytical approach leads to the prediction of reasonable
values for the lift coefficient, but to get better agreement with
experimental data, an empirical correction factor is introduced.

The investigation, conducted at the Experimental Towing Tank,
Stevens institute of Technology, Hoboken, N. J., is the thesis sub-
mitted by the author in partial fulfillment of the requirements for
the degree of Master of Science.

NOTATION

TT Geometric angle of trim

X Wetted length-beam ratio (— )

b

C^ Speed coefficient (Froude Number) V/Vgb

g Acceleration of gravity

b Beam of planing surface or span of airfoil

V Velocity of the free stream

C^ Lift coefficient, r —

^ ipV^bi

L Lift force

r Fluid density

jZ Wetted length of planing surface or chord of airfoil

m

S Projected wing area, I I cfydx

F Normal force **

1 b^
Aspect ratio, r- or ^

■ If-

0( Ambient flow angle of attack

q/. Induced angle of attack

O Spray thickness

V. Induced downwash velocity infinitely far aft

lac

m Virtual mass rate of flow, VpVol.

KE Kinetic energy

/I^KE Change in kinetic energy

r Wetted length measured from trailing edge to stagnation
line on planing surface

A O.ht^

B '^^ h sinT

A

^~- Einpirical wave rise factor

r-^ Wetted length-beam ratio neglecting wave rise

Vi/Vz Snpirical correction factor

Notation Used in the Section on the Effect of Gravity

Velocity potential

U Free stream velocity

5, '7 Coordinates of point (x,y) on a planing surface

(x,y,z) Coordinates of a point in space

10 Wave number ^p

U

IjL Coefficient introduced by Maruo to obtain desired value

^ of certain integrals

P (f,"*7) Pressure at any point on planing surface

9 Wave direction angle

k General wave number

w Downwash as used by Maruo

1^ A function of Froude Number

B Buoyancy force (Archi median Force)

Notation in Appendix 1

^ Geometric angle of attack

oC Ambient flow angle of attack
e

w^ Induced angle of attack

w Complex potential

Velocity potential

Y stream function

z Complex variable, z = x + iy

T Kinetic energy

S Surface

V Transverse flow velocity

Notation in Appendix II

R Force

R Drag force

X

Q Volume flow in spray jet, Q = V^

06 Angle of trim

I^ Spray resistance

D Total lift f orce, dynamic plus Archiniedian

R^ Frictional Force
tp

R^ Wave resistance

V A function of Froude Number^ = r-

2 2 Fr

2 V

Fr Froude Number squared 5 —q

M Moment

C Lift coefficient (C )

•'''d Center of pressure location from trailing edge

P, S, R Vector forces

h Depth of fluid from free surface

THE LIFT COEFFICIENT OF FLAT PLAINING SURFACES

TOTR.nTITnTTnN

Ifydrodynamic phenomena arising from planing on the surface of
water have aroused considerable interest in seaplane hull^ float, and
hydroski enthusiasts. In the past the need for immediate design data
for floats or hydroskis has led, based on many specific experimental
investigations, to the development of numerous empirical formulae to
describe the flat plate lift coefficient as it varies with other para-
meters such as angle of trim^ 'V, wetted length-beam ratio, X, and
Froude Number, V/vgb, While these empirical formulae agree very well
with experimental results within the range of values for which they
were developed^ the validity of such formulae outside the range of data
for which they were developed becomes unreliable or at least question-
able.

The objective of this work is to develop on a rational basis,
insofar as possible, a fonmila for the lift coefficient of a flat
plate which planes on the surface of water. At present, a rationally
derived formula developed by H„ Wagner, Ref . 1, covers the infinite
span case, neglecting gravity, while the more recent work of L„ I. Sedov,
Ref, 2, and H, Maruo, Ref, 3, covers the infinite span including the
effect of fluid gravity which involves wavemaking as well as hydrostatic

effects. Methods of correction for the effect of a finite span neg-
lecting gravity have been developed by such writers as B. V. Korvin-
Kroukovsky, Byrne Perry, W. Bollay, W. G. A. Perring and L. Johnston,
W. Sottorfy L. Landweber, F. W. S. Locke, and G. L. Shuford, Jr.,
Ref. ii-12, but the entire process, including the effects of trim,
aspect ratio, and Froude Nximber, has not been integrated except in
a preliminary form.

This work involves an investigation of the methods and applic-
ability of derivations previously made by the investigators of flat-
plate planiQg. The problem of deriving a formula on a rational basis
is treated in two distinct parts. First, the aspect ratio, trim angle
effect^ neglecting gravity, is considered; then, second, the effects of
gravity including wavemaking are discussed.

To extend the range of experimental data for comparison with re-
sults developed analytically, an experimental investigation was made
which is described herein.

The investigation has been conducted by Lto D., D. Farshing, Jr.,
U.SoN., a Naval Postgraduate School student at the Experimental Towing
Tank, Stevens Institute of Technology, Hoboken, New Jersey. Acknow-
ledgement is made to Professor B. V. Korvin-Kroukovsky of the Exper-
imental Towing Tank for his helpful guidance and interest in this work.
Gratitude is also expressed to the members of the Towing Tank staff Yiho
have willingly given their assistance and advice throughout the course
of this investigation.

MODIFICATION OF THE TWO-DIMENSIONAL PLANING SOLUTION
FOR THE EFFECT OF A FINITE SPAN IN A WEIGHTLESS FLUID

Historical Background

There have been several attempts to find closed form solutions
to the finite span planing surface problan, but difficulty arises in
that formulation of the problem to satisfy the boundary conditions
introduces a non-linear integral equation. Solution of the integral
equation is feasible only after many simplifying assumptions have been
made.

To circumvent the difficulty arising from the integral equation,
another approach is offered. An ideal weightless fluid is assimied to
exist. Based on the method of potential flow, a two-dimensional solu-
tion is found, then modified for the effect of a finite span by a tech-
nique similar to that used for correcting the two-dimensional airfoil
to finite spano

A brief but comprehensive review of the significant work in plan-
ing up to 19h9 has been presented by William Siler, Ref . 13 • Charles
L. Shuford, Jr., Ref« 12, has given a recapitulation of the results of
many investigators who sought to find a simple formula for the three-
dimensional lift coefficient of the planing flat plate. After his re-
view of the attempts of other investigators to find simple formulas,
Shuford proposes another method for predicting the lift coefficient.
Shuford observes that experimental data indicates a non-linear rela-
tionship between the angle of trim and the planing lift coefficient

for the planing surface of finite span. To account for the non-
linearity, he approaches the problem with an analogy to the low aspect
ratio airfoil. Appendix II shows further justification for this ap-
proach. To arrive at a final total lift formula, Shuford combines
three components; the first, a linear term given as one -half the value
of the flat plate airfoil lift coefficient based on the lifting line

theory; the second, a non-linear, cross flow term which is shown to be

2
proportional to sin ^f', and the thirds a suction component of lift

existing at the leading edge of the airfoil but not present in the

planing surface flow.

Shuford' s total lift formula becomes

„ r/2 mr ,. . 2 X „ 2_

L ^ 1-^PB. ~ • ~ ^"^ ^' * ^^ ^°^ '^
values of which agree well with experimental data„ The analysis by
Shuford, while in good agreement with the available experimental data,
is not completely theoretical since the planing problem was not solved,
but, instead, the airfoil problem manipulated.

In many of the papers concerned with modification of the two-
dimensional solution for the effect of finite span, methods similar to
those used for correcting the infinite span airfoil have been used with
varying degrees of success „ The technique of adopting similar methods
seems to be a logical approach since the flow is assumed ideal (without
friction) and weightless.

Another successful approach is one made by Byrne Perry, Ref . 6.
Perry has applied the low aspect ratio airfoil analysis of Fo Weinig,
Ref o lii, to the low aspect ratio planing surface. Weinig solved the

airfoil problem without recourse to the two-dimensional solution. He
applied the momentum theorem to a control surface around the airfoil
with virtual mass and induced angle of attack as independent variables.
The virtual mass used by Weinig is that associated with the lifting
line plus a wedge of fluid found by the projection of wetted length
normal to the flow:

J sin tr. (See Fig. 1)
Appendix I presents more in detail the concept of virtual mass and the
work of Weinig insofar as it is understood. In order to justify the
choice of virtual mass made by Weinig, the applicable work of M. M.
Munk, Ref . 1^, is also discussed. The formulation of the low aspect
ratio airfoil problem and its solution by Mlliam Bollay, Ref. 16, is
considered appropriate for discussion in Appendix I, and it is com-
pared with the solution obtained by Weinig,

The principal objection to the planing surface analysis by anal-
ogy with the low aspect ratio airfoil of Weinig is that the method for
obtaining the induced angle of attack is not clear.

The Two-Dim ens ional Solution

H, Wagner, whose applicable work on planing is found in Ref. 1,
obtained a two-dimensional solution to the planing problem. As pre-
viously stated, an inviscid, weightless fluid is assumed to exist.
Coordinate axes are fixed in the planing surface which has infinite
span and infinite forward length. Fluid is allowed to flow toward
the plate parallel to the x axis shown in Fig. 2. The mass of fluid

6.

of thickness % contained between the free surface and the stagnation
streamline as shoivn on Fig. 2 is deflected forward along the plate.
The fluid below the stagnation streamline continues aft and finally
becomes part of the wake. With the aid of the mathanatics of complex
variables using conformal transformations, Wagner finds the normal
force per unit width of the infinite span to be:

^ 1 + cosr- (1 - cost)Jln ^=^~,i-Trsiiir

2 cos'Cr

The projection of this hydrodynamic force perpendicular to the direc-
tion of flow gives the lift force per unit span. Then:

L = ^Ov^ 27r sin t:" cost::'

^ 1 + cosr- (1 - cosrUn 1-cosg ^^^^

2 cosr

or

L _ 2 7r'sin rcos r

^^*'P^ " 1 H- cosr- (1 - cosT;)ien lz22||^rsinr
For the infinite span case, 'r =^d and o^,. = 0.

Application of the Momentum Theorem - Method I

Let us now take the infinite span solution of Wagner and modify
it for the effect of finite span.

To begin we must put a control surface around the finite span
planing surface so that the momentum theorem can be applied. For the
control surface let us choose a solid shape of infinite spanwise
width. One face is infinitely far ahead of the planing surface; one
infinitely far aft; one at infinite depth below; and the remaining
face at the free surface, a^bjCjdjejf of Fig. 2.

(1)

7.

There is no momentum flux through the side and lower faces of
the control surface, so we can focus our attention of the fore and aft
faces and that part of the free surface face which cuts through the
forward spray.

Infinitely far ahead of the planing body, the flow is uniform
and parallel to the undisturbed free surface, while infinitely far
aft, a constant downwash, v,.^, is achieved. The downwash concept in
airfoil theory is well established and is permissible in an ideal fluid
without gravity. In the real fluid, frictional forces as well as grav-
ity wave fonnation invalidate the discussion of a downwash infinitely
far behind; however, what happens physically behind the airfoil or
planing surface is not important as long as the mathematical formula-
tion of the phenomena occurring at the body is mechanically compatible
with the assumptions made about the fluid.

With the established concept of a downwash infinitely far aft of
the planing body, and no downwash infinitely far ahead, it can be
seen that there will be a change of momentum in the negative Z direc-
tion. Momentum flux resulting from the forward spray occurs at the
control surface cutting through the spray, c d on Fig. 2. The time rate
of total momentum change gives a force in the Z direction, and this
force on the fluid is applied by the planing body.

Let us now consider the rectangular flat surface of finite beam
planing at velocity V. If we fix axes in the planing body, the flow
picture is that shown in Fig, 2. The flow about the plate can be re-
solved into two components, viz,, a component along the plate and one

8.

transverse to the plate. It is the transverse component of the flow
which for the moment becomes important. The transverse component of
flow gives rise to a virtual mass described by iifegner to be the mass
of a semi-circular cylinder of fluid. The virtual mass is not a
physically existing mass of fluid but it does take on physical signi-
ficance. Appendix I describes how the virtual mass concept comes in-
to existence J but for the present 5 suffice it to say that Wagner has
found "Dhe virtual mass associated with the transverse component of
flow about the planing body to be one-half that associated with the
lifting line. M. M, Munk, Ref . 15 j shews how one finds the virtual
mass of the lifting line by association with the potential flow solu-
tion of transverse flow about a line immersed in the fluid. Munk's
work is further discussed in Appendix I, Extension of the transverse
flow about a line to the case of the planing plate is almost trivial,
and the results lead to the mass of a semi-circular cylinder of
diameter equal to the span.

The force of the plate on the fluid results from the time rate
of change of the product of the virtual mass and an appropriate velo-
city. Similar to Weinig's low aspect ratio airfoil theory, the virtual
mass associated with large values of wetted length-beam ratio planing
surfaces becomes the mass of the serai-circular cylinder plus the mass
of fluid found by j^sinr where Ji is wetted length and t the angle of
trim (see Fig. l).

Justification for using the virtual mass similar to that used by
Weinig, and justification for describing the virtual mass as Weinig

9.

has done, may be found in Appendix I.

From the above analysis, we find the time rate of change of

momentum from the inlet face to outlet face of the control surface

to be:

F„ = m v.^^- vertical component of spray momentum (2)

where _

m =(^ +Ab sinr)?V

One should note that the F„ in equation (2) is the force of the plan-
ing body directed vertically down (in the negative Z direction). To
find the actual lift on the planing body, F„, it is necessary to
eliminate the spray force. Let us define m as the mass rate of

flow in the spray. Then

m = ??bS
s

where 5 is the spray thickness (see Fig. 2),

The lift equation by application of the momentum theorem then
becomes:

L = (^ + Jib sinr) €Vv^^- €VTdS V sim? (3)

If equation (3) is divided by 2V we have

O xr

L ^TT /7i^ nv- . 't i*" ^VbSsinr' n\

2v =^^ ^T" * ^^ ^^'^^ 2V 2V ^^^

Transposing the spray term of equation (ii) to the left side, then

dividing by the coefficient of v. /2V we have:

2 V
L - gy b S sin V i^

2 ~ 2V

267^ (22^ + j^b sirn^

(^)

Defining the length beam ratio as:

E = '^

and the lift coefficient, C,. as:

c, = ^

equation (5) becomes:

' |?v2bA

Z * Zl/x, sinr V

It now becomes necessary to discuss v^^2V. At a great distance ahead
of the planing body, we found the flow parallel to the free surface or,
in other words, the downwash velocity zero (v. =0). At the center
of lift of the planing body, we assume the downwash to be some frac-
tional part of the final downwash velocity infinitely far aft of the
bocfy. For the moment, let us call the downwash velocity at the bocfy

k V. , where k is a fraction. From the momentum equation (neglecting
loo

spray), the force on the bocfy is:

Fe
= -m V. _,

and the work per unit time done by the bocfy on the fluid

Work = F k V.

The kinetic energy per unit time associated with the downwash in-

finitely far aft is:

T>"ri m 2
KE = -x V.

2 i<»

while at infinite distance ahead it is zero. The change in kinetic

energy is equivalent to the work done so that:

ATv-T-, m 2 o , 2
Kb = 7yv. =mkv.

From this analysis, it is evident that k must equal l/2 so that at
the center of lift of the planing body the downwash is one-half its
value at an infinite distance afto Now it should be clear that v,. /2V
is the slope of the flow at the center of lift of the planing surface,
and the angle whose tangent is v,. ^2V is called the induced angle of
attack or induced angle of trim. Symbolically:

V.

tan U ^ = pTp

Equation (6) now becomes:

C^ + 2 f sirnr

-^ ^ = tanO«^. (7)

(^ + h sinr) "

An inspection of Fig. 1 shows the relationships between the induced
angle, 0C_. , the geometric trim angle, 7^, and the ambient flow angle, 6^ .

r = 0^ +<j6. (8)

e 1

Let us now return attention to equation (1) which is the two-
dimensional solution of the planing flat plate as developed by
H, Wagner. The angle of trim "C" in the two-dimensional case is
equivalent to the ambient flow angle., oC^ since the induced angle
is zero when gravity is not considered. Replacing 'V by oi in equa-
tion (1) and assuming oi, sufficiently small^ the following approxima-

tions can be made:

cos 0^ « 1

1-cosft^

is fractional and its natural logarithm is negative.

2 COS06

e

1-coset

-(l-coso^ )il.n 7J y~ is positive but very small.

e c cosofr
e

12.

The denominator of (1) then becomes*

1 + coso^^ 2
and the lift coefficient becomes:

2 +ysinot

e

Now it becomes necessary to perform some algebraic manipulation

upon equation (9).

C 2 7rsinoi

L e

2 " h * 25}^sin o^
e

C, - 2 2 'yaa:fi oZ. -li-2r9ino*
L _ ^ e e

2 h + 2jfs±n oL

C. - 2 - -^

2 + jrsin o^g

TTsin 0^3 = cTTT^ - 2

-Ck+ 2 Cj^ -X)

s^ <^e = ~rTcp2i

^^ <^e = 7^(2 - C^)
0^ = arc sin

e - '^'^ """ 7r'(2 - G^)

Expanding the arc sin in series:

2C^ 8Cj^^

For values of C^ ^ «85j the cubic term is less than 3% of -the first
term and subsequent teims are much smaller. Neglecting the cubic and
all subsequent teims of the series we have:

13.

From Ref . 1 an expression for the spray thickness is given as
follows :

s X

I

1 ■♦• cosy * 1 - cosT^ ^ -ysin y
1 - cosi?"'^'^ 2 cost/ 1 - cosi

t! *"" 2 COST/ 1 - cost/
Referring to Fig. 2, one notes that r is measured from the trailing
edge to the stagnation point while J, is measured from the trailing
edge to a line tangent to the spray root. The difference between r
and J? is considered insignificant. If in equation (?) we now let
tanoi:; » U. , equation (8 ) becomes

2 G,

r =

2 9rsinr

c 1 + cosr_

^ 1 - cos -C

^^ 1 - costr . rsinr
^^ 2 COS 7^ 1 - cosT^

n{2 - o^)

The following substitutions are made to simplify the algebra:

2Vsin'C

Let

1 + cos

1 - COS

r .^ 1 - cos7> ^ T/sinTf
-€ "* 2 cos tr 1 - cos'

= A

(11)

(12)

and ^^ * ^ ^^"''^ " ^

Equation (12) then takes the fonn:

tr =

2C.

lf{2 - C, ) *

B

Clearing fractions and collecting like powers of C^ we find:

c^ - (Br+ ie + 2 - k)Q^ _ 2(A - B-r) =

and solution by the quadratic formula yields:

^

B('fc'+|) + 2 - A

B f+ ^ B + 2 - A

+ 2(A - BT^

(33)

(lli)

To facilitate numerical calculations, one notes that A can be simpli-
fied considerably, Ref. 1 contains a numerically evaluated plot of

%/r vso TTo Fran zero to about h^ degrees, the curvature is slight so
that a straight line appiDximation will not introduce material error
in subsequent computations » The linearization is carried out by making
the straight line coincide with the curve at zero and 2^ degrees. The
slope K of the straight line is then found per radian to be approximately
0.2 so that:

A = 2 J- sin r*
becomes approximately

A = (2) (0<,2)^2 = Oolfl? (15)

Values of A and B are tabulated in Table I, then the quadratic equation
is solved o The values found from solution of equation (lii) without
linearizing S/j^ and those found when equation (1^) was substituted
for A show no appreciable difference; therefore, the values of A in
Table I are those found using equation (l5)o Table II contains the
numerical values of equation (lit) 3 and Fig,, 3 is a plot of these values.

Modification of the two-dimensional flat-plate-airfoil solution
for the low aspect ratio case by the method used here led to results
which, compared with experimental data^ overestimated the lift. Simi-
larlyj the planing results developed here overestimate experimental
results.

Application of the Momentum Theorem - Method II

The fact that planing lift results are overestimated by t^he above
development leads one rather logically to the conclusion that the vir-
tual mass chosen was too large Reflection upon the phenomena occur-
ring at the plate then leads one to conclude that a new definition of

1^.

virtual mass based upon the ambient flow angle rather than the geo-
metric angle of trim should be used.

Redefining the virtual mass 5 we obtain

L = ^VbA(^ + sin e<'^)v^^- ^V^b gsin OL^ (I6)

Dividing by 2V, transposing, and simplifying as before, there results:

^i-" . . C^^ . 2 I sin 0^^

•^p = tan U. = — —

(^ + li sino|)

(17)

Substituting for sin 0^^ the value found from the two-dimensional
solution, we find:

C. + 2

%

2C,

0^. =

^ I 1-^(2 - C^)

\

l^^

y(2 - C^)
Again substituting in equation (8) there is obtained;

2C,

r =

2C,

+ -

2^-^Cl ^ ^ e^L

(18)

2\ 2 r- yc

L

Simplification of equation (I8) using the linearization for l/jl as
previously described leads to the cubic equation:

,3.

[4

nr hY^rr 1 ^ 16 ,

If 2\ Tf

r'

4(¥^-^^|)^4^^

c,-2_%o

(19)

Substitution of X = in equation (I9) recovers the two-dimensional
results.

16.

Solutions for equation (19) are tabulated in Table III and
plotted on fig. Iio

The values of C, obtained by this method underpredict the ex-
perimental values above values of X of about 1.^ to 2, One may con-
clude then that the virtual mass selected in this manner is too small
for the low aspect ratio case and too high for the large aspect ratio
case. The low aspect ratio airfoil treatment by this method also
underestimates the lift.

EXPERIMENTAL INVESTIGATION

Purpose

A comparison of experimental data with the results computed by
the two formulas for the lift coefficient developed herein was under^
taken. It soon became evident that existing published data did not
cover the entire aspect ratio-angle of trim range desired; therefore,
the necessary tests were performed in the towing tank to supply the
otherwise unavilable data. To this end, the facilities of the sea-
plane tank at the Experimental Towing Tanky Stevens Institute of
Technology were employed.

Equipment and Procedure

The model used for the tests was machined from half-hard brass
bar stocks and provision for supporting the model in the towing car-
riage was made by mechanical linkages shown in Fi^. 5" Th© overall
length of the model was 18 inches, the beam 2 inches ^ giving the

17.

general appearance of a rectangular ski. The lengthwise edges of the
upper surface of the ski were beveled at an angle of 30 degrees as
shown in the sketch of Fig, 5. The upper surface bevel provided
sharp edges for a clean break-away of flow. The flat bottom which
was the wetted surface during the tests was polished and marked with
Dykem Steel Bluing to permit ease of reading the wetted lengths frcm
the \inderwater photographs made during each test run. The marking de-
tails are also shown in Fig. 5. To eliminate the adverse effects of
spray on the carriage, it was found that vertical separation between
the model and carriage supporting linkages was necessary. To provide
the required vertical separation, two mounting rods, approximately
3-l/i; inches long were tapped into the upper surface of the model
near the after end.

The seaplane tank where the tests were conducted is 313 feet
long, 12 feet wide, and 6 feet deep when the water level is just even
with the opening in the standpipe inside the tank. A maximum carriage
speed of 50 feet per second can be attained with the selected speed,
amplidyne controlled, carriage driving motor. A tachometer generator,
the armature of which is driven by the carriage driving motor shaft,
supplies electrical infonnation to the field of the amplidyne system
to provide speed control. With a selected voltage, representing a
certain speed, the carriage speed can be maintained to within O.06
feet per second. A General Electric photoelectric tachometer records
the carriage speed.

18,

The camera for making underwater photographs of the wetted sur-
face was a specially adapted Beattie Varitron, Model E, Data Record-
ing Camera. The Varitron is a low speed, pulse type camera, the operar-
tion of which is completely automatic. An electrical signal operates
the shutter of the camera, and as the shutter opens, it triggers the
photoflood lights to make the exposure. Following the exposure, the
shutter is automatically disconnected while another circuit provides
power for winding the film in the magazine to the next frame. When
film winding is completed, the shutter circuit is re-engaged, and the
apparatus is ready to make another exposure. A recording chamber on
the side of the camera automatically records sequence numbers and in-
formation on a platten to identify the test. The camera for the tests
described herein was housed in a watertight box and pressurized to in-
sure against leakage. A control for re3:Qote focusing after the camera
had been positioned in the tank was also providedo A light source,
mirror, photocell, and thyratron were used to actuate a relay which
delivered the electrical pulse to the camera shutter.

The model was installed in the overhead towing carriage as shown
in Fig. 5 with provisions made to set the angle of trim as desired.
With the trim angle set at zero degrees, adjustments were made to in-
sure that there was no angle of roll or yawo By gently lowering the
ski into the water and observing the bottom surface with a mirror held
underwater below the ski, a simultaneous wetting of the entire bottom
surface indicated proper alignment in rollo law was checked by plac-
ing a bar with reference marks inscribed thereon at the front edge of

the ski, perpendicular to the direction of motion and in the plane of
the bottom surface of the ski. Then, if by moving the carriage for-
ward, the edge of the ski remained on the reference mark, yaw angle
was considered zero. A spirit level protractor was used to set the
angle of trim.

With the desired angle of trim set, a calibration in heave was
made so that during the runs, a first order approximation on the
wetted lengths could be ascertained. To accomplish the heave cali-
bration, the ski was again gently lowered into the water and observa-
tions in the underwater mirror were made similar to those for the roll
adjustment. When the trailing edge of the ski just touched the water,
the heave reading was taken on the large scale as shown in Fig. 5»
This reading was listed on the data sheet (Table IV of this report)
as the heave reference, and this reference value was subtracted from
the heave reading made during the run. The difference (or sum of ab-
solute values in the case of a negative heave reference) gave the
vertical projection of the wetted length of ski excluding the wave
rise. Then by an empirical wave rise factor, described in Ref . 18,
the approximate total wetted length could be found. Let it be empha-
sized here that the heave readings were used in the tests as a control
measure only. Since there are wave disturbances in the tank after the
first run, slight transient effects limit the accuracy of heave read-
ings. The procedure for obtaining approximate wetted lengths from
heave readings was as follows (see Fig« 6):

Let: -g- = wetted length/beam ratio of the ski neglecting
wave rise
h = difference in heave (mnning heave minus reference

heave, inches)
^^ * empirical wave rise factor = <,3
"C = trim angle (degrees)
X = J^ = total wetted length/beam ratio

b b b b sinf I'

^ = b = rilrt * -3 h = (X - .3Xb Sim;)

Anticipating specified wetted lengths made it is possible to
determine approximately the desired value of h so that during the
runs, the values of h served as a check on actual wetted lengths
being obtained. Adjustment of the load to get the desired value of
h and, thus, the actual wetted length, was then practical.

In order to eliminate all effects of gravity on the lift co-
efficient obtained by the tests, all runs were made with a speed co-
efficient, Cy, at a nominal value of twelve. Theoretical justifica-
tion for independence of gravitational effects on the total lift coeff-
icient at high Froude Numbers will be treated later, (Refer
to Fig. 7)0 Since the beam of the model was two inches, a carriage
speed of 27»8 feet per second was found to be appropriate. (Note that
C^ = V/Vi5 or V = Cy \/gB).

For each run, a pre-determined load was applied to the model
which was allowed to run free in heave with a fixed angle of trim and

at constant speed » The wetted length for each run was photographed
so that correlation with load could be made for data reduction.

Table IV, which is a copy of the laboratory data sheet, contains
all of the information pertinent to each of the test runs. Table V,
containing wave rise "correction" factors computed from the heave
readings and the underwater photographs, was made as a check on the
previous empirical wave rise factor and is intended to supplement
present existing information for future tests which other investiga-
tors may perform in the aspect-ratio-angle of trim range covered
herein.

Experimental Results

Reduction of experimental data from the photographs taken and
carriage speeds recorded are found in Table VI,

Since most of the experimental data collected from the tests
here were b^ond the range of existing data, it was not possible to
evaluate the reproducibility of such results, particularly in the
limited time availableo The general trend of curves faired through
the test points obtained in this investigation is to lie between the
curves of Fig. 3 and k which leads one to believe that the results are
reasonable. It is noted from the ratios of computed values based ono^
to experimental values tabulated in Table VII and shown on Fig, 8 that
the data collapses fairly well for trim angles of 18 degrees or less.
Above 18 the dependence of experimental values of the lift coeffi-
cient upon the angle of trim manifests itself, but it is not clear

22,

from a physical viewpoint just what the relationship between \, 7^,
and C, should be.

GQLPARISQM OF THEORETICAL RESULTS WITH EXPERE/IENTAL DATA

General Discussion

In order to detennine the validity of calculations made by cor-
recting the two-dimensional planing solution for the effect of finite
span, a comparison was made with experimental data. The principal
source of data used was from Weinstein and Kapryan, Ref. 19^ Since
the existing data available did not cover the complete angle of trim,
wetted length-beam ratio range desired, additional data were obtained
experimentally and are presented herein. Of the several methods avail-
able for comparing data, it was decided that ratios of computed values
to experimental values plotted against X with fas parameter would
give the most enlightening results. Values of the ratios using both
method 1 and method 2 previously described for computing values of
C. are found in Table VII and are plotted on Fig. 8 and 9« Since the
ratios are not of the order of unity, it becomes necessary to make an
empirical correction to the computed values in order to predict the
lift coefficient satisfactorily.

Empirical Correction to the Curves of C, Computed by the Cubic Equation

For m defined as a function of 0^ , it is seen that as X becomes

e'

greater than about 2, computed values of G^ from equation (19) imder-
estimate the experimental values. The ratios of computed C^ to ex-

23.

perimental Cj values found in Table VIII and plotted on Fig, 8 show
that for trim angles of 18 degrees or less, the ratio plot tends to
collapse into a single curve. For the case of C- computed on the basis
of equation (lli) the plot of ratios, Fig, 9j does not collapse as neat-
ly as the plot of Fig. 8, For this reason, empirical correction to the
curves of G, computed by equation (19) was elected.

The curve shorni on Fig. 8 was faired to fit most of the points
and the empirical factor found as follows:

L computed

Q

L experimental _ , /„ -,

K

where yi = empirical correction factor.

For 2°^ 7:'^l8°, >^ = f (X) and we call this factor y^^. For
18°^ 7:^^30°, >(= t(\,lf) and this factor is denoted by yi^.

There are, of course, several suitable functions to fulfill the
requirements on h, but the ones chosen here are:

\ = 1,3^9 - tanh (^^) (21)

7^2 ="^1 + i^ ^anh X^ (22)

From Fig. 8, the intercept value of the curve is recovered from >^,

when X = 0.

o _ 18°
In equation (22), Al^is defined as ' ^^„ — radians. Then when

\- 0, \= \,

^he empirical factors, Y), and >7p, have been applied to values of

C- computed by equation (19) and the results tabulated in Table VIII,

Fig, 10 is a plot of the empirically corrected values of Ct and is

21,.

the one to be used for estimating the lift coefficient when gravity-
is not considered.

Fig. 11 shows the empirically corrected curves of Ct vs. X with
experimental data points plotted thereon.

THE EFFECT OF GRAVITY.

One can recognize that the influence of gravity in the case
of a planing body can manifest itself in two separate ways. One
part of the gravity contribution is a static force which is the
Archimedian lift or buoyancy; the other is a cfynamic effect which
attends the phenomena of the development of waves.

Until recently, the effects of gravity on the dynamic lift have
been practically ignored in planing lift calculations, partially be-
cause the mechanism of the dynamic effects has not been clearly under-
stood (Ref. S, 17, 20), and partially because the theory, including
the dynamic effects, is quite complex and difficult to evaluate.

In 1937, L, I. Sedov, Ref. 2, presented a two-dimensional asymp-
totic solution to the planing problem, including the effects of
gravity on the dynamic lift in the region of high Froude numbers.
For more practical applications, Hajime Maruo, Ref. 3, published in
1951 a two-dimensional solution to the planing problem, and followed
this in 1953 J Ref. 21, with a solution for the finite span.

It is considered appropriate to summarize here the Maruo paper,
Ref. 21, which reveals the effects of gravity on the lift of the
planing flat plate of finite span.

25.

Llaruo assumes a form for the velocity potential function ^ which
involves an unknown pressure distribution over the planing surface.

o

This potential function is constructed to satisfy the free surface

condition of constant pressure except on the planing surface. Then,
from the potential function, a general formula for the downv/ash is
found. _« •© ■)

8 -rr o

Maruo points out that the downwash can be separated into two com-
ponents; one corresponds to a profile characteristic and is present
even in the two-dimensional flow; the other is a downwash arising
from the effect of finite span.

The downwash condition applied at the planing surface to satisfy
the condition of no flow through the plate, gives rise to an integral
equation for the unknown pressure distribution over the plate.

Maruo, by a procedure analogous to Glauert's solution of the
finite span airfoil problem, simplifies and solves the integral equa-
tion for values of X ^1/2, or for aspect ratios greater than about
two. Figo 7 is a graphical representation of the solution when X— ^0.

One should note on Figo 7 that there are two curves, I and II,
Curve I is the plot of a numerical solution to the exact equation
while II is a plot of the simplified equation evaluated analytically.
Attention is directed to the shape of the curve of Fig, 7 which shows

26.

that the theoretical value of C^ , not considering buoyancy effects,
rises from a C^ of 0.7^ and approaches asymptotically a constant value

at higher C„. Most plots of experimental data show the total lift co-

1 2
efficient without subtracting the coefficient of buoyancy 3/^ ^V b /

from the data^ therefore, the effect of gravity on the dynamic lift
appeals to be a monotonically decreasing function of ^„ rather than
an increasing function as shown by the theory.

The practical results of Maruo's work are given by Equation (1^2)
of Ref . 21 which, when transcribed to the notation used herein, be-
comes :

where ]^ is a function of the speed coefficient, C„,

A graph of the T^ function, given in Ref. 21, can be linearized
to the following approximation:

^= 0,667 + %^.

Then C^ becomes

'Y

If

\ " 0.667>^ ^^ 0.333 + Q.233>^ ^^^^

G 2

From equation (23), it is evident that the lift coefficient becomes
independent of the cfynamic effects of gravity for

. / 0.7X

v^ yi -^ 2x

One should be aware of the fact that the mathematical validity of
equation (23) was based on \ <rl/2o

27.

For small X (such that 2X<-< 1), it is seen that the first tenn
of the binomial expansion of the inequality yields C„> 0.6V^which
agrees quite well with the empirical value of C /V^>1 established
by Daniel Savitsky, Ref. 18, as a criteria for independence of gravity
in the flow.

The effect of gravity on the dynamic lift from this analysis is
seen to become negligible at low values of Cy. Those investigators
who insist that the value of G„ above perhaps 10 is necessary to elim-
inate the gravity effect refer to the combined buoyant and dynamic

effects. The dynamic effects become small at low values of C^., but

2
until the dynamic pressure, l/2oV , becomes large enough to make the

ratio of static buoyant force to dynamic pressure a small percentage

of the total lift, buoyant effects will persist.

For most practical cases, the lift coefficient, as Cy approaches

the order of 10, is clearly independent of all gravity effects. If

one subtracts from the total lift coefficient the buoyant lift con-

B
tribution defined by T~Z „2, p, it is seen that the remainder is in-
dependent of all effects of gravity only for C > 0.6V5vrwhen X ^ 1/2.

SUmyLARY AND CONCLUSIONS

The results of this investigation show that theoretical evalua-
tion of the effect of a finite span on the planing surface problem is
quite complex. To remove some of the complexity, the problem was con-
sidered in two phases. First, the effects of gravity were neglected,
and an ideal fluid assumed; then second, the effects of gravity were
discussed.

28.

Neglecting gravity, equation (ll|) was obtained from a combination
of the two-dimensional solution and application of the momentum
theorem to a selected virtual mass. Equation (19) was found in a
similar manner but with a different virtual mass. Both equations (lli)
and (19) overestimate the lift coefficient for aspect ratios greater
than about one-half which indicates that the choice of virtual mass
for that range of aspect ratios is too high. Equation (I9) underesti-
mates the lift when wetted length-beam ratios are greater than about
two (aspect ratio less than one-half), while equation (lii) overesti-
mates the lift for all length-beam ratios.

Fig. 1 shows cross-sectional areas of the "volumes" containing
the virtual mass for each case. Tables II and III give tabulated
values of the calculations made by each of the above methods, and
Table VIII gives the empirical corrections to make values computed
by equation (I9) agree with experimental data. JEkpirically corrected
values of the lift coefficient are plotted on Fig. 10, and comparison
of these curves with experimental data are shown on Fig. 11.

While the analytical results obtained are considered reasonable,
empirical correction to the theoretical analysis was required to get
better agreement with experimental data. Thus, the development can be
considered as a semi-empirical method for evaluating the effect of a
finite span on the solution to the planing problem in a weightless
fluid.

For the analysis made independent of gravity at length-beam-
ratios greater than about 2, the method leading to equation (19)

29,

indicates that the virtual mass may be too small. The virtual mass
associated with equation (lii) may be too large. The most prominent
physical explanation for the inconsistency between theoretical and
experimental results is that the downwash along the wetted length of
the body is not of constant magnitude. Bollay, Ref. 16, indicates
that free vortex lines are shed along the chord of the low aspect
ratio airfoil so that at the trailing edge^the contribution to down-
wash tnere is not only from the vortex wake, but also from the shed
vorticity ahead. This, in effect, produces a curvature of the flow
from leading to trailing edge which has the effect of introducing an
apparent camber to the foil.

For the low aspect ratio (large \) planing surface, the curvature
effect may have a significant influence on the lift, and is offered as
a possible explanation for the difference between predicted and experi-
mental values .

It is worthy of notice here that the technique similar to that
used for modifying the infinite span airfoil for the effect of finite
span is a logical procedure to use for the planing surface. As
pointed out by many other investigators, the flow for most practical
purposes can be considered ideal and weighless in the planing regime,
and from the stagnation point to trailing edge on the lower surface,
both airfoil and planing body have almost identical pressure distrib-
utions.

The consideration of gravity led to results which show that the
effect of gravity on dynamic lift is a function of the speed coefficient

30.

and the wetted lenpth-beam ratio. It was found that for all practical
purposes, the effect of gravity on dynamic lift could be disregarded
for G ;p^ 0.6V5rwhen \'<Zl/2. Buoyant effects remain significant
until the dynamic pressure reaches such a value as to make the co-
efficient of buoyancy very small compared with the dynamic lift
coefficient.

31.

BIBLIOGRAPHY

1. Korvln-Kroukovskyj B. V. : Hydrodynamic Design of Seaplanes II,
Experimental Towing Tank, Stevens Institute of Technology,
Hoboken, N.J. (Summary of the applicable work of H. liVagner
included ) .

2. Sedov, Lolo- Two-Dimensional Problem of Planing on the Sur-
face of a Heavy Fluid, Unpublished Translation by B.V. Korvin-
Kroukovsky, Experimental Towing Tank, Stevens Institute of
Technology, Hoboken, N.J., 1937.

3. Maruo, Hajime: Two-Dimensional Theory of the Hydroplane, Pro-
ceedings of the First Japan National Conference for Applied
Mechanics, 195l»

ii. Korvin-Kroukovsky, B.V. : Lift of Planing Surfaces, Journal of
Aeronautical Science, (Readers' Forum), Vol. 17, No. 9j September
1950, pp. 597-599.

5. Perry, Byrne: The Effect of Aspect Ratio on the Lift of Flat
Planing Surfaces, Report No. E-2ii<,5 (Contract N6onr-2lili2li,
Project NR 23J4-OOI), Hydrocfynamics Lab., California institute
of Technoloi^r, September 1952 o

6. Perry, Byrne: The Lift of Flat Planing Surfaces, Preliminary
Report No. N-3I.35 Hydrodynamics Laboratory, California institute
of Technology, March 1950.

7. Bollay, William: A Contribution to the Theory of Planing Sur-
faces, Proceedings Fifth Int. Cong. Appl. Mech. (Cambridge, Mass.,
1938), John Wiley and Sons, Inc, ^ 1939^ pp. hlh-hn '

8. Perring, W.G.A. and Johnston, L. ; The Hydrodynamic Forces and
Moments on Simple Planing Surfaces, and an Analysis of the ifydro-
dynamic Forces and Moments on a Flying-Boat Hull, R & M No. I6J46,
British Aeronautical Research Committee, 1935*

9. Sottorf , W. : Analysis of Experimental Investigations of the
Planinp Process on the Surface of iVater, NACA T.M. No. I06I, I9I1I1.

10. Landweber, L. and Eisenberg, P.: Characteristic Guides for
Planing Surfaces, David Taylor Model Basin Report R-80, 191^3 .

11. Locke, FoW.S., Jr.: An Bnpirical Study of Low Aspect Ratio
Lifting Surfaces with Particular Regard to Planing Craft,
NAVASR DR Report No. 10ii3, January 19li8.

32,

12, Shuford, C,L. , Jr.: A Review of Planing Theory and Experiment
with a Theoretical Study of Pure Planing Lift of Rectangular
Flat Plates, NACA T„N. 3233, Langlqy Aeronautical Laboratory,
August 195^0

13. Siler, William: Lift and Moment of Flat Rectangular Low Aspect
Ratio Lifting Surfaces, Experimental Towing Tank Technical
Memorandum No, ^(:>^ Unpublished, 19ii9»

Ih. Weinig, F. : Lift and Drag of Wings with Small Span, NACA T.M.
No. 1151, August 19ii7.

15 c Munk, Max M, : Fundamentals of Fluid Dynamics for Aircraft
Designers, The Ronald Press Co,, New lork, 1929.

16. Bollay, William: A Theory for Rectangular Wings of Small Aspect
Ratios, Journal of Aeronautical Sciences, Vol. IV, No. 7, May
1937.

17. Weinig, F. : On the Theory of Hydrofoils and Planing Surfaces,
NACA T.Ivi. No, 81i^, January I938.

18. Savitsky, Daniel and Neidinger, Joseph W, : Wetted Area and
Center of Pressure of Planing Surfaces at Very Low Speed Co-
efficients, Experimental Towing Tank, Stevens Institute of
Technology, a Sherman M, Fairchild Publication Fund Paper by

the Institute of Aeronautical Sciences, Deptember 195^j No. FF-11.

19. Weinstein, Irving and Kapryan, Walter J, : The High Speed
Planing Characteristics of a Rectangular Flat Plate Over a
Wide Range of Trim and Wetted Length, NACA T,N. 2981, Langley
Aeronautical Laboratory, July 1953 •

20. Wagner, H. : Planing of Water Craft, NACA T,Mo No, II39,
April 191480

21. MaruOj, Ha jime : Theory of the Hydroplane with Broad and Short
Planing Surfaces, Bulletin of the Faculty of Engineering,
Yokohama National University, Yokohama, Japan, Marcy 1953»

22. Milne- Thorns on, L.M, : Theoretical Aerodynamics, MacMillian Co.,
Inc., New York, May 191^7.

23. Perelmuter, A. ^ On the Determination of the Take-Off Character-
istics of a Seaplane, NACA T.M. No. 863, May 1938.

2[i, Winter, H. : Flow Phenomena on Plates and Airfoils of Short Span,
NACA T.M, 798, 1936,

33.

2\$. Lawrence, H.R. : The Lift Distribution on Low Aspect Ratio
Wings at Subsonic Speeds, Journal of Aeronautical Sciences,
Vol. 18, No. 10, October 19^1.

3ii.

APPENDIX I

General

The use of incompressible potential flow theory to treat the
effects of low aspect ratio on wing lift has been carried out for
the case of a flat plate airfoil in at least two distinctily different
ways, but with essentially the same result. The two different methods
cited are:

(1) The solution by F. Weinig, Ref . lli, whereby the momentum theorem
is applied with an estimate of virtual mass, and an estimate for
the induced angle of attack, ^^, is made based on cascade theory,

(2) The integral equation solution of W. Bollay, Ref. 36, in which
the boundary condition — that there be no flow through the
plate — is satisfied by equating the transverse velocity com-
ponent of the stream, V sino", to an integral expression estab-
lished for the downwash at the wing.

It should be noted that both of these methods are developed for
an airfoil in an incompressible fluid neglecting gravity; therefore,
it is reasonable to expect the results to be analogous to that por-
tion of the planing problem which is treated independent of gravity.
Wagner, Perlemuter, Perry, and Shuford, Ref. 1, ^5 6, 12, 20, and 23,
have indicated that the integral of pressures over the wetted surface
of the planing body, excluding the region about the leading edge
forward of the stagnation point, is almost identical with the pres-
sure distribution over the same portion of a flat plate airfoil;

35.

thus, the planing surface-airfoil analogy seems to be reasonably-
founded.

It is not the purpose of this discussion to present the details
of the airfoil solutions of Weinig and Bollay, but rather to show
how they lead to a better understanding of the virtual mass chosen
to modify the two-dimensional planing surface solution for the effect
of a finite span. The fact that the solutions of both Weinig and
Bollay lead to essentially the same result is graphically revealed
in Fig. 12. The experimental points shown in Fig. 12 were obtained
by H. Winter, Ref. 2I4, and are included to show the relative merit
of each of the theoretical results displayed. One obvious reason for
the difference between the theoretical solutions of Weinig and Bollay
is that mathematical complexity forced Bollay to make several simpli-
fying assumptions in order to solve the integral equation established
in his fonnulation of the problem. Although Bollay' s treatment of
this problem seems to be established on a more rational foundation,
Weinig' s results correlate better with experimental data as pointed
out by HoR. Lawrence, Ref. 2^.

Virtual Mass

Modification of the two-dimensional planing surface problem for
the effect of a finite span by application of the momentum theorem
makes necessary the choice of an appropriate virtual mass. Attention
is directed to the low aspect ratio airfoil solution by YiTeinig wherein
the virtual mass selected is composed of the virtual mass associated

36.

with a lifting line plus the streamwise projection of the chord
bJcsin oC„ In order to better \mderstand Weinig's choice of virtual
mass, a review of the salient features of the virtual mass concept
as given by MoM. Munk, Ref. 1^, are considered. A brief review of
the most important steps leading to the definition of virtual mass
follows .

If an incompressible fluid at rest is given an impulse which re-
sults from the action of normal pressures alone (frictional forces
assumed zero) the resulting flow is potential flow provided no dis-
continuities exist and all spacewise partial derivatives of the poten-
tial function are continuous » If it is assumed that the impulsive
force creating the flow acted for some finite distance, work would be
done on the fluid, and since the fluid was originally at rest with
zero kinetic energy, the work done represents the kinetic energy
stored in the flowo It is convenient to replace the total energy
of the fluid particles moving in many different directions with the
equivalent energy of a certain geometrically defined mass of the fluid
moving with the velocity of the bodyo This assumed mass, introduced
for mathematical convenience, is labeled virtual mass and may be
different in magnitude for different directions,, depending on the
velocity chosen in its definition.

If a very thin airfoil with axes fixed in the body moves with
a steady velocity in air which was originally at rest, velocity about
the airfoil at angle of attack,©^, can be resolved into two components,
one nomal and one parallel to the foil as shown in the sketch.

37.

In an ideal fluid (no friction),
the component parallel to the air-
foil does not contribute to the
forces. Attention is then concen-
trated on the normal component which
is referred to as the transverse
flow component,, The line pq on the sketch represents an element of
the airfoil of span bo (It has been shown that when the span to chord
ratio is large, the entire foil may be represented by such a line,)
The wake at some distance behind this airfoil behaves like a flat plate
of span b moving with velocity v perpendicular to the plane described
by the plate. The complex potential for the flew net about this hypo-
thetical flat plate is givai by:

w - -^ iyr - iv (z - \lz^ - (|}^) (Al-1)

where z = x ■*■ iyj v = transverse velocity component^ and b/2 = semi-
span. The product, p0» can be thought of as the impulsive pressure
necessary to create the flow field in which kinetic energy is storedo
Then, the expression for kinetic energy in terms of the potential func-
tion is :

sj oefS^ds

where O = fluid density

d0/dn = component of transverse flow velocity
ds ~ surface element over which the integral is performedc
This is the total kinetic energy of the flow, and as previously

(Al-2)

38.

stated, it can be expressed in terms of the virtual mass, m, as

T = 1/2 mv^ (Al-3)

where v = d0/dn = constant on the surface of the plate. The total
kinetic energy of the flow as foiind by the potential function is
equated to the energy expressed by the virtual mass so that

Since o = constant and d{^/dnj = v = constant, there is obtained:

Jplate

vol = ij ^S (Al-5)

This defines the volume of fluid of density p associated with the
transverse velocity. It should be noted that the virtual mass is
not a physical body of particles; its properties ^ as stated previously,
are directional such that for a direction other than that of the trans-
verse flow, the virtual mass may be entirely different.

On the surface of the h^rpothetical plate (i.e. y = in the com-
plex variable notation):

■•vP-

= V Yl - x'=^ (Al-6)

Substituting the value of from equation (Al-6) into equation (Al-5),
and noting that djMn is constant over the plate equal to v, the
volume of fluid defining the virtual mass is obtained:

Vol = y I - x^ dx (Al-7)

Since the surface over which the integral is performed extends
from -b/2 to +b/2 on both upper and lower surfaces of the plate, the

final expression for the volume becomes

/b/2 r~2 ' 2

r/| - x^ dx =-^ (Al-8)

-b/2

39.

This is the volume of fluid containing the virtual mass and is shown
here to correspond to the mass of a circular cylinder of diameter
equal to the span of the plate.

The Lifting Line

In the previous derivation of lift over the span b and an element
of length dJ?, the downwash velocity, v = d0^/dn was necessarily con-
stant along the span and was assumed to have no chordwise variation.
The same constant downwash velocity is known to exist in the case of
the vortex system consisting of bound and trailing vortices such that
the circulation of the bound vortex varies elliptically along the
span. The bound vortex is known as the Lifting Line, and it can be
taken to represent the entire wing when the ratio of span to chord
is large.

For multiple lifting lines disposed vertically in close proximity
to each other, Munk has indicated that the interference existing be-
tween the flow patterns associated with each line reduces the virtual
mass of the single lineo The mathematical complexity involved for
the case of three or more lines limits the discussion here to only
qualitative, order of magnitude results.

In the case of a biplane, two lifting lines separated by a small
vertical distance, h, can be assumed to replace the wings. The re-
sulting volume containing the virtual mass associated with the trans-
verse flow is given by Munk as approximately the cylinder of diameter
equal to the span plus the rectangular mass enclosed between the upper

and lower wings: _

Vol =2^ + bh

The Effect of a Long Chord

The virtual mass associated with the lifting line, used in
application of the momentum theorem to find the lift for a finite
span airfoil leads to the prediction of reasonable values for span
to chord ratios which are large. As the chord lengthens with respect
to the span, the single lifting line representation becomes invalid,
principally due to the fact that the free vortex lines no longer
leave the trailing edge in a flat plane of the main stream, but in-
stead are shed along the chord forward of the trailing edge to form
a three-dimensional vortex array in the wake.

In order to satisfy the boiindaiy conditions d0/dn = V sino^
on the surface of the airfoil, and of the streamlines flowing smooth-
ly off the trailing ed^e, Bollay, Ref. 16,22, has assumed a distribu-
tion of bound vortex lines along the chords After certain simplifi-
cations, Bollay has solved the integral equation for the distribution
of circulation among these vortex lines (or lifting lines).

Interpreting Bollay 's results in terms of virtual mass, the
chordwise spacing of ar^r pair of bound vortex lines at a distance d*
apart leads to an equivalent vertical spacing of d£sino(.. Consider-
ing this in the light of Munk's theory of the biplane and multiple
lifting lines, the integral along the chord of the contributions of
all vortex pairs gives the contribution bJcsino6to the volume of the
virtual mass as an approximation. The total volume associated with

ia.\

an airfoil of small span and long chord then becomes "t — + bfcsinoi.
It is observed that this is the exact expression chosen by Weinig.
It should be noted, however, that od in this case needs an inter-
pretation,, Weinig has assumed it to be equal to the geometric angle
of attack of the airfoil, but in the spirit of the above derivation,
it would appear more logical to use the ambient flow angle in the
immediate vicinity of the airfoil rather than the geometric angle of
flow.

Modification of the two-dimensional airfoil solution, using the
momentum theorem, with the virtual mass based on geometric angle of
attack, then on the ambient flow angle, follows.

Modification of the Two-Dimens ional Airfoil Lift Coefficient for the
Effect of FinJte Span

Method 1 - virtual mass based on geometric angle of attack,©^.
From the momentum theorem:

oyfot

L = A V. = &^ (f^ X sino<)v. (Al-9)

where m = (rj~ + b/^sinodj^:^

b
From energy relationships:

V.

7~ = tan£^^(V„ (Al-10)

Dividing (Al-9) by 2V, substituting the result of (Al-lO) and solving

forO^^, , there is obtained

" (2V)(eVb.£)(f + X sino^/ TT^ UK sino,^

QC, = ^^i— — - = ---_il^_-^ (Al-11)

a2.

where C, =

^ le^^u

Now:

Q a.

and from two-dimensional theory:

\ = 2Trsino^^ co^^
Dividing (AI-I3) byTT:

— r = 2 sin(y coso^ = sin2o^
Jf e e e

2^ = arc sin —
e 77-

Vi^ritinp the series expansion for arc sin

TT-

*o = l^

I • &r i •

Neglecting all terms in the series higher than the cubic, and sub-
stituting from (Al-16) and (Al-11) into (Al-12):

2 I 77" [jij 6 ( 7r+ ^^ sinQ6

Solving :

.3 1^ , 125^

C^ - 12WoC=

'L '!"" ■ 77+ hX sino6| L

Values of Q^ substituted into (AI-I8) show no appreciable dif-
ference from the results obtained as follov^s.
In equation (AI-I3) let:

Then (AI-I3) becomes:

cos oi :::i 1
e

sinO^ ^^
e e

L e

(Al-12)
(Al-13)

(Al-lit)
(Al-15)

(Al-16)

(Al-17)

(Al-18)

(Al-19)

ii3.

Multiplying (Al-12) by 27/7 then substituting from (Al-11) and (Al-19),
there is obtained:

2 7rxc

2iroi = C + ■ ^ |. ■. — , (Al-20)

Equation (Al-20) simplifies to:

TT + ijX sinoi
Values of C_ computed by (Al-21) are tabulated in Table IX and plotted
on Fig. 13. It is observed that this method overestimates the ex-
perimental values of Winter,

Method 2 - virtual mass based on ambient flow angle, o^ .
Similar to method 1, there is obtained forO^. :

xc ■^•

Oi. = ^^ 1..^ ..^^ (Al-22)

jr ■¥ \\K sino^

Approximating again:

cos 0^ ~ 1
e

sinO^ fi.0^

\ e e

and substituting©^ = 'orr^ ^^® equation:

e 1
there is obtained:

Simplifying and collecting coefficients of like powers of G :

■^.■r

=l' 'm * '"'' - ^'^K - '^ = (A1-2W

hh.

Equation (Al-2[i) is solved by the quadratic formula, values tabulated
in Table IX and plotted on Fig. I3 for comparison with method 1 and
the experimental data of Winter. It is observed that C^ is underesti-
mated by equation (Al-2l^).

APPMDK H
Foundations of the Theory of Planing

Translation of Paragraph 75 of the Russian Textbook, Water Resistance
to Movement of Ships, by P„ A. Apuchtin and J, I. Voitkunsky,
Moscow, 19\$3»

The viscosity of a fluid affects the pressure distribution along
the planing surface very little and separation of the boundary layer
is not observed on planing surfaces. This provides the basis for in-
vestigation of the phenomena of planing by methods of hydrodynamics of
a perfect fluid, in particular the potential flows.

In the domain of the theoretical investigation of planing, the
Soviet scientists played a leading part. In 1929, G. E. Pavlenko
first obtained the solution for the lifting force and resistance of
a plate of infinite span, planing at a small angle of trim on the sur-
face of a perfect fluid.

Later, G, E. Pavlenko developed the foundations of the theory of
planing, taking into account the finite span^ studied the physical
phenomena connected with planing, and also investigated the question
of recalculation of model test results of planing surfaces to full
size.

In subsequent works, the two-dimensional problem of planing of
plates on the surface of a perfect fluid was attacked along two paths:

1) by the methods of streamline flows, not taking into account
the weight of the fluid.

hs.

2) by methods of the theory of waves of small height taking into
account the weight of the fluid.

The investigations of planing by means of streamline flows for
infinite depth and for finite depth of a fluid were conducted by the
academicians S. A, Chaplygin, Mo I. Gurievich, A. P. Yampolsky and
Wagner.

The solutions of the planing problem of a curved plate, taking
into account the weight of the fluid were given by the academicians
N, E. Kotchin and L. I. Sedov.

The planing of the surfaces moving one after another (in tandem),
interesting in connection with the action of steps, for the case of
gravity-less fluid was investigated by L. I. Sedov.

Let us consider the origin and the calculation of water resist-
ance for the planing flat plate on the surface of a perfect fluid.

Large pressure gradients are observed along the lower surface in
the movement of the plate. Plots of the pressure distribution,
measured experimentally along the centerline of the plate and along
its edges at various angles of trim, are shown on Figo I3J4. Along
the span of the plate pressure changes are not significant, while
normal to the span in the region of the leading edge the pressure
gradient is very large<, A large drop of pressure is observed at the
lateral edges, where the pressure abruptly drops to a value corres-
ponding to the pressure in the undisturbed fluid.

in regions of large pressure gradient (in the regions of inter-
section with the free water surface), spray jets are formed. The pro-

(1) See Figo lii

hi.

jection of the reaction of the jet, in the direction opposite to the
direction of the plate movement, represents spray resistance.

Let us consider the flow of a parallel stream of velocity V onto
a stationary plate. Consider the depth of the fluid limited, equal to
h. The influence of gravity on the flow of the fluid will be neglected.

Flow of the fluid in such a motion is steady. Applying Bernoulli's
equation to the particles on the free surface (Fig. 135)} and taking
into account the fact that pressure on the free surface is constant,
it can be shown that the velocity along the surface of the jet is con-
stant also, and is equal to V.

In the actual motion of the plate, the velocity in the jet is
double that of the plate motion. Isolate by means of a control surface
the closed volume of the fluid ABCDEF. ' According to the law of
momenta, the flow of momenta through the control surface is equal to
the vector of external forces applied to the volume, ABCDEF,

The difference of momenta flow through the sections of the con-
trol surface, AB and CD, is equal to VQ, where Q is the mass flow in
the spray jet.

Projecting the flow of momenta through the control surface ABCDEF
and the vector of forces applied to the plate on the axis OX, we get

\ = pVQ (l+cosO(j)
and the total reaction is directed normal to the plate and is equal to

R . 3L- = fVQ ^^^ = pVQ ctg^
sinot V sin 06 \ 2

(1) See Figc li|

148.

These formulae are valid for any value of the angle o^; the theory
discussed can be considered therefore as the non-linear theory of
planing.

As the fluid velocity in the spray jet in the case of a stationary
plate is equal to V, the volumetric flow is

Q = V^
where ^ is the thickness of the sprayo

The force, R , in the case considered is equal to the spray re-
sistance, R^ of the plate. Finally, the formulae for computation of
the spray resistance and total hydroc^namic reaction in planing on the
surface of a gravity-less fluid of arbitrary length at any angle of
trim will take the following form:

H ' P^^^ (I+COS06) (136)

R=^V^^ctgf (137)

If the plate of infinite span planes on the surface of a fluid,
located in the field of gravity, waves are formed in its wake. In
this case, the additional force of wave resistance will act on the
plate.

In the case of the motion of a planing plate on the surface of
a perfect fluid, therefore, the total resistance is the sum of spray-
making resistance, Rr, and of the wave resistance, Rg, Considering
formula (135 )^ in the light of the above derivationj we find that
Rg + R c = D tano^«

(1) \ = D tanod+ R^p (135)

D = total lift, ic eo dynamic + Archimedian

R. = the force of friction
tp

1^9.

The sum of forces Rg + Rr represents the projection of the re-
sultant force of normal pressures, distributed along the plate, onto
the direction of motion, i.e. represents resistance due to pressures.

With the growt.h of relative velocity, the part played by the
gravity forces diminishes in comparison with inertial forces; there-
fore the part played by wavemaking resistance in the total resistance
of the plate diminishes.

The comparison of the relative importance of the wavemaking and

spray resistance in the total resistance of the planing of the flat

(1)
plate is given on Figure I360 ^

The complicated calculations permitted L, I. Sedov to obtain the
formula for computation of the ordinates of the free fluid surface,
hydrodynamic pressure, lifting force, resistance and the hydrodynamic
mcment acting on the planing surface.

In eases of large relative velocities (small values of the para-
meter V= 2 Y ^ ^^ ^^^ formula for calculation of the lifting force, re-
sistance, and hydrodynamic moment of the flat plate were obtained by
means of Sedov' s theory;

R = R^ =pTT(i -TTv^ - l494-)a M^oC (I38)

R^ = Rod (139)

M = 2r (1 „ §1^) a^vl^ (11,0)

where a = tt and F = V/\/gt.

These formulae show that at the large relative velocities and
small angles,©^ 5 the lifting force and hydrodynamic mcment in planing

(1) See Figo Hi

so.

are linear functions of the angle of trim.

More accurate numerical calculations of the lifting force and
moment by Sedov' s theory were made by Uo S, Chaplygin for a large
variation in the parameter ,V • Results of these calculations are
compared on Figo 137 with results computed by formulae (I38) and
(IhO). On Fig. 137(a) and (b) are plotted the dependence of C,/cC
and the relative distance of the center of pressure from the rear
edge of the plate '^^/J? on the number Fr« = v/VgJ?. Ccanparison of the
curves shows that beginning with Fr, = 2<.8, i.e. withV= 0.06iij the
results of the refined numerical computations and of calculations by
approximate formula coincide« In the case of very large Froude
numbers, Fr., it can be assumed that>^= 0; this assumption means
physically that the gravity of the fluid is not considered. With
the introduction of such an assumption, the formulae for the flat plate
are further simplified to:

R = a^ = pJTa V^ (liil)

R^ = pTia V«^ (lii2):

M = |pTra^V?>^ (IU3)

Were the distance of the center of pressure from the trailing
edge of the plate computed by means of the above formula for the
moment, it would be found that jK , = (3/li)2a (i.e. m,^ = ol^)°, this
agrees well with test results.

Let us compare formula (lijl) and (lii2) with the formulae for the
computation of corresponding quantities obtained in the wing theory
for a flat plate of infinite span in a streamline flow at small angles

(1) See Fig. lit

51.

of attacks

R^ = 2pTra y^c6 (llili)

M =^77'a^V^od (lli5)

The flat plate represents the simplest wing. The comparison of these
formulae shows that in planing at large Froude numbers the lifting
force and moment of the plate is half that obtained in the flow
around the plate deeply submerged in a fluid. This relationship
points to the analogy of the flow on the lower part of the plate
in both cases.

The development of the analogy between the planing surface and
the wing has permitted application of a number of relationships ob-
tained in wing theory to stucfy the phenomena of planing.

From the theory of the wing it is known that in a flow about a
plate at an angle of attack 0^5 in accordance with the theoran of
Joiikowskyj a lifting force occurs j normal to the direction of the
flow at infinity o The velocity at the trailing edge of the plate
has a finite value, and the streamlines flow off smoothly. At the
sharp leading edge the velocity tends to become inf inite^ and in
connection with this 3 the pressure in the vicinity of the leading
edge drops sharply, ttydrodynamic pressures are normal to the surface
of the plate, and their resultant, P, is also directed along the
normal to the plate (Fig. 138,a),^ ' 3h addition to the force P, a
suction force, S, acts in a direction along the plate. The total
hydrodynamic reaction^, the lifting force of Joukowsky, R, is the
vector sum of forces P and S;

(1) See Fig. 1^

52.

R = P + S

The analogy of the flow in the space under the place in the
case of planing at small angles of attack, and in the case of the
plate fully immersed in a fluid, is valid from the trailing edge to
close proximity of the leading edge. Analogous also are pressure
distributions in this part of both flows.

The flow is different only in the region of close proximity to
the leading edge where the spray jet directed forward occurs along the
plate (Fig. laajb).^-"-^

From the above reasoning, the results of the study of flow about
a plate fully immersed in a fluid can be used for the study of flow
under the planing surface if the flow about the leading edge (where
velocities grow to infinity) is replaced by streamline flow reproduc-
ing the spray jet. Such a replacement at small angles of trim will
cause but an immaterial change in the area of the noraml pressure plot
diagram at the lower edge of the plate. Taking into account the fact
that the flow in planing occupies only the lower part of the space, it
can be expected that the resultant of the normal force R at the planing
surface is equal toi

R =7/2.
The validity of this relationship was already noted earlier.

Developing the analogy, we can suppose that the reaction of the
spray jet is directed in the opposite sense to the force of suction S,
and in magnitude is equal to R, = 3/2. Making use of formula (I36)
for the spray resistance in the case of planing on the surface of a

(1) See Fig. lU

^3.

gravity-less fluid

Rj^ = Spv (1+cosC^)^
and letting cosOd= 1 for small angles,06 and TL = S/2, we can compute
the thickness of the spray jet:

where S = suction force of the wing and is given as:

S = 2?7pa Vot^

The validity of the above analogy is conf inned experimentally at
large Froude numbers up to values of the angle, 0^, from 7 to 10 de-
grees.

In the case of flat plate planing at an arbitrary angle of trim,
the resultant of hydrodynamic forces, R, can be calculated by
foiTOula (137)

R = pV^J ctg ^

The curve K = f(pOs showing the ratio of the magnitude of force R,
calculated by formula (137) to the magnitudes calculated by formula (llil),
(i.ea on assumption of small angles 0^)j is plotted on Fig, 139<«
Fig. 139 shows that with increase of the angles^, the magnitxide of
the resultant hydrodynamic pressure, computed by non-linear theory,
becomes smaller than that computed by linear theory, i.e. the analogy
between the planing s-arface and a wing is disturbed.

The theory of planing permits one to appraise the effect of
shallow water and of limited channel width on the hydrodynamic charac-
teristics of planing surfaces.

(1) See Fig. li^

^h.

The detailed investigation of the effects of shallow water on
planing upon a gravity-less fluid at various angles of trim was made
by U. S. Chaplygin.

As a result of these investigations it is established that, in
planing on shallow water at small angles of attack, the analogy with
the wing moving between parallel walls is observed. Theoretical cal-
culations show that hydrodynamic characteristics in motion on shallow
water depend on the ratiOytrh, and the angleo^, where h is the water
depth.

With increase of the ratio j2:h, i.e. with a decrease of water
depth, the lifting force at constant velocity increases.

On Fig. liiO, ^ ^ curves are given permitting one to estimate the
increase of the lifting forces caused by shallowness of water at various
values of the parameter X/:h and angles of trim,0<^.

Taking into consideration the effect of channel walls on planing
without consideration of gravity indicates that, at the ratio of
channel width to the width of the plate b:B <.7, hydrocfynamic forces
acting on the plate increase^ at bsB^- 7 the effect of walls leads to
a small (less than 2%) reduction of forces as compared to motion on
the surface of unbounded fluid.

(1) See Fig. Ik

55,

PQ

CO

^1

CD

O

o

o

•sl^

•\

c e

O .H

•H Ji

-P Eh

OS

+j <;h

3 o

&-

e <s>

o ^

O M

c

^. <5

o

<M o

M

•H

CO ^

s

-p +3

G 0)

m

<l> 6

<!

•H O

en

O 0)

•H O

<Vh

<« C

a> o

o

O tJ

ID

<*-i M

O OS

m

a

o 'S

•H

4J M

03 G

r-l .H

3 w

,o D

CO

o

o
tl

OQ

-<

CD

CO
EH

IfJ

'^

to

C\J

PQ

PQ

^

S^

•H

CO

1— IC^lO<-HrHirDtom

tocvii-HcnifiaiOco
0'i<oocx)cna>CDCD

•-tCMWinC-i-HlOCn
oooooooo

CMCMCVJCMCMtOtOlO

CO
CO

COCJOCDCOOOOO
lOa)eOTtlTJ^l/^■^i^l— I

c~-oooco5i<oocvi<o

oooooooo
CMCJtOlOCOtO^^

CO

iH

CO

o

CVJ

coraOcocooOO

COCOLOC\JC0CO00C~-

COO'^inuDCOtOCM

OCMtOlOf-fHLOOJ

oooooooo

C\J
o

to

CDCV30«3«30000
tOtOC\3a5ir500'*

ascot^c-cooicoui

a)rHC0';t<COO';t<00
oooooooo

c-c»c»oocoa>oi05

CO

CDCVJOCDCOOOOO

cna>com.-icDCDO

COt-fHcvjtOCOCMO
^-^C^J■5i^CDC0(^0CDO

oooooooo
i-H i-H CM

a>com'#o^oc-0
^a5T}<coc--cr5Coo

a

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CD

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r-t

1—1

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°

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co

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UD

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■"^

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3 OO'-t<-HCMC0r}<U5

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cocdlocmoocococ*-
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56,

TABLE II
Compiutation of C. Using m Based on Geometric Angle of TrinijT

^L ■

2/?r) + 2 - A
2

V

3(1: +

2/it) * 2 - aT^

1
2(A - BZ)

-

2 J

1 2

3

B(r+2/fr)-^2-A

2

k

Of

5

2(A-Bi:)

6

7

X Z'

V®*©'

^L

).2 2

3.683

13.561I

-.557

3.606

.077

h

3.871

111. 985

-1,131

3.722

.lli9

6

It. 061;

16.516

-1.723

3.81i6

.218

9

ii.360

19.010

-2.61ili

a.0li5

.315

12

ii.665

21.762

-3.602

h.26l

.ItOii

18

5.301

28.101

-5.633

l;.7ii3

.558

21^

5.968

35.617

-7.803

5.27li

,69k

30

6.661

lilj.369

-10 100

5.853

.808

).li 2

2.365

5.593

-,283

2.30I1

.061

k

2Mh

6.170

-.583

2.363

.121

6

2.608

6,802

-o901

2.ii29

.179

9

2.801

7.81i6

-l.ii30

2.537

.26I1

12

3.00ii

9.02li

-1.958

2.658

.3I16

18

3.ii3ii

11,792

-3.165

2.937

.lt97

2k

3.887

15.109

-li.500

3.256

.631

30

ii.383

19.210

-5.988

3.635

.7li8

).6 2

1.925

3.706

-.191

1.875

.050

li

2.022

li,088

-.hoo

1,920

.102

6

2.123

li,507

-.627

1.970

.153

9

2.282

5.208

-•>999

2.052

.230

12

2.I150

6,003

-l.lilO

3.II12

.308

18

2.812

7.907

-2.3ii3

2.359

.li53

2k

3.210

10.30I1

-3.1il6

2.62li

.586

30

3.62U

13.133

.-1;.617

2.918

.706

).8 2

1.705

2e907

-,ll;l

1.662

.01^3

k

1.791

3.208

".309

1.703

.088

6

1,880

3.5311

-,li90

1.7U5

.135

9

2.022

li,088

-.793

1.815

,207

12

2,173

1^.722

-1.135

1.893

.280

18

2,501

6.255

-1.932

2.079

.li22

2k

2.860

8.180

-2,868

2.305

.555

30

3.2iili

10,52li

-3.931

2.568

.676

57.

1.0

2.0

k.o

6.0

8.0

TABLE II (Cont'd.)

2

3

B(j;+2/7r)+2-A
2

li

Of

5

2(A-Br)

6

7

r

\p^d

^L

2

1.57ii

2,ii77

-.118

1.536

.038

h

1.6\$2

2,729

-.25I1

1.573

.079

6

1.735

3.010

-.1(07

1.612

.123

9

1.866

3.1i82

-.670

1.677

.189

12

2.007

a. 028

-.971

1.7^8

.259

18

2.31ii

5.355

-1.685

I.9I6

.398

2lt

2.6^2

7.033

-2.539

2.120

.532

30

3.016

9.096

-3.520

2.361

.655

2

1.310

1.716

-<,o6U

1.285

.025

h

1.375

1.891

-.lii5

1.321

.o5h

6

lo ilUii

2.085

-.2ii3

1.357

.087

9

1.555

2.i4l8

-.1^23

1.1^12

.lli3

12

1.675

2.806

-.61^2

l.ii71

.20I4

18

1.9iil

3.767

-1.191

1.605

.336

2i4

2,238

5.009

-1.881

1,768

.li70

30

2.561

6.559

-2.698

1.965

.596

2

1.178

1.388

-,036

1.163

.015

it

1.236

1,528

-.090

1.199

.037

6

lo298

1,685

-.161

1.23ii

.061i

9

1.399

1.957

-.300

1.287

.112

12

1.509

2,277

-.ii78

1.31a

.168

18

1.751i

3.077

-,9kh

1.1^60

.29li

2ii

2,031

I4.125

-1.552

I.60I1

.li27

30

2.333

5.to

-2.287

1.776

.557

2

I.I3I1

1,286

-.027

1.122

.012

h

1.190

l.lil6

-.071

1.160

.030

6

1.250

1.563

-.133

1.196

.05I1

9

I.3I47

l,8lii

-.259

1.2li7

.100

12

l.i;53

2,111

-.ii23

1.299

.151^

18

1.692

2.863

-.862

l.lilli

.278

2li

1.961

3.8ii6

-l,iii42

1.550

.liii

30

2.257

5.09ii

-2.150

1.715

.51^2

2

1.112

1.236

-.022

1.102

.010

li

1.167

1.362

-.062

l.liiO

.027

6

1.225

1.501

-.120

1.175

.050

9

1.321

1.7li5

-.238

1.227

.0911

12

1,1|26

2.033

-.395

1.280

.lii6

18

1,661

2.759

-.821

1.392

.269

2li

1.927

3.713

-1.387

1.525

.li02

30

2,219

h>92k

-2.081

1.686

.533

TABLE II (Cont'd.)

58,

1

2

3

B(t+2/?r)+2-A
2

h

5

2(A-Btr)

6

7

X

V®*©'

^L

.0

2

1.099

1.208

-.020

1.089

.010

h

1.153

1.329

-.057

1.128

.025

6

1.211

I.I166

-.111

I.I6I1

.Oli7

9

1.305

1.703

-.226

1.215

.090

12

l.li09

1.985

-.379

1.267

.lii2

18

1.6ii2

2.696

-.796

1.378

.26ii

2ii

1.906

3.633

-1.355

1.509

.397

30

2.196

a, 822

-2.0U0

1.668

.528

59,

TABLE III

Computation of Lift Coefficient Usinp a Virtual Mass, m
Based on Effective Anf le of Attack, Qt

X

'=L=

4- (2.292 - -

^)r-3.

''' - 41 'l'

"2

.4*(^f'

- 4.584) t

6.283r

2

3

4

5

6

r

deg

Coeff of

Coeff of

Const

Solution

P

q

r

^L

2

-3„354

6.059

-.219

,036

4

-3„329

6.119

-.439

.074

6

-3 .304

6.178

-.658

.113

9

-3.266

6.267

-.987

.166

12

-3.228

6.356

-1.316

.232

18

-3.152

6,534

-1.974

,350

24

-3.077

6.712

-2.631

.482

30

-3 .002

6.889

-3.288

.610

2

-2.826

4.950

-.110

,022

4

-2.774

4.900

-.219

.045

6

-2.721

4,850

-.329

,070

9

-2.642

4.774

-.494

.110

12

-2.563

4.698

-.658

.151

18

-2 .406

4.547

-.987

,250

24

-2.248

4.397

-1.316

,360

30

-2 .090

4.245

-1,644

.480

2

-2.563

4.395

- ,055..

.012

4

-2.496

4.290

-.110

.026

6

-2 ,,430

4.185

-.164

.040

9

-2.331

4.02 7

-.247

.064

12

-2.231

3.870

-.329

,090

18

-2.032

3.553

-.494

.155

24

-1.833

3.238

-.658

.237

30

-1.635

2.923

-.822

.330

2

-2.475

4,210

-.03 7

.009

4

-2 .404

4.086

-,073

.017

6

-2.333

3.963

-.110

,028

9

-2.227

3.777

-.164

.044

12

-2.121

3.592

-.219

.060

18

-1.908

3,222

-.329

,110

24

-1.696

2.852

-.438

.170

30

-1.484

2.481

-.548

.240

60.

1

K

10

TABLE nK Cont'd.)

i

3

4

5

6

r

deg

CoefC of

Coeff of

Const

Solution

V

^

r

=L

2

-2,431

4.118

-.02 7

.007

4

-2,358

3,985

-.055

,014

6

-2,285

3,852

-,082

.021

9

-2.175

3,656

-.123

.033

12

-2,065

3,454

-,164

.049

18

-1.845

3,056

-.247

.085

24

-1,626

2.659

-.329

.134

30

-1.407

2.261

-.411

.200

2

-2 ,404

4,062

-,022

.005

4

-2,330

3,924

-,044

.011

6

-2,255

3,786

-.066

.018

9

-2,144

3,579

-,099

,029

12

-2,032

3,372

-,132

.040

18

-1,808

2,957

-,197

,069

24

-1,585

2,543

-,263

olio

SO

-1.362

2,129

-.329

.160

<;-.
<«

»o

^O

X

§

1^

-P

-p

•fi

o

c-

CO

^

(D

o

p

■ — ■

CO

-3

1

C\)

>> >>

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tn U

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u.

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Cm <m

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to

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CO

a> a>

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CO

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t. t<

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Jh

03 OS

-p

4)

CO

PL, &,

(0

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CO

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to

,— 1

<D

e

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o

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CO

V«j^

>>

(X,

u

c-

ix

tt

CO

o

+3

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CO

e

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8

61

<M CM CO to ■^

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CJOtOOtOtnyCDLDOOlDOOOtomO
t~-CV3C>JcoCMC£)tD(X>C0O)CDcO^tOcO

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rHrHrHiHrHiHiH iHiHrHCOCOCO COCO

62

TABLE V

Wave Rise "Correction" Factors for Wetted Length Determinations

from Heave Readings

b =

2"

Angle

Photo

Veloc ity

Speed

Wetted

Heave

h^

Wave Rise

of

Noo

V

Coeff.

Length /Beam

(Reading

sinT^

Factor

Trim

(ft/sec. )

C

Ratio

Min\is

(X-h^ sintr)

IT

V

from Photo

Ref.)

(deg.)

X

K

51i9

27.

.72

11,96

2,30

1,17

1.89

.Ix^

5^0

27.

.72

11,96

2.30

1.17

1.89

.hi

^51

27.

.78

11.99

5.07

2.77

IM

.58

/%

552

27.78

11,99

5.05

2.77

ii.li9

.56

18°

553

27.

.72

11,96

6.55

3.70

60 00

.\$^

55U

27.

.72

11.96

6.I13

3.67

5.95

.I48

555

27.78

11.99

7.22

a.23

6.85

.37

556

27.78

11.99

l.hh

ii.37

7.08

.36

557

27.

.72

11.96

2o68

1,80

2.22

.1^6

558

27.

.72

11,96

2.63

1,78

2.19

oUi

559

27.

.72

11,96

5.87

ii.25

5.23

.e\x

560

27.

.72

11.96

5,82

U.20

5.16

.66

2U°

561

27.

.66

11.9ii

7.25

5.33

6.55

.70

562

27.

.72

11.96

7.12

5.25

6.1^5

.67

563

27.

.72

11.96

3.80

2.66

3.27

.1^3

56ii

27.

.72

11,96

3.80

2.65

3,26

.5ii

^e^

27.72

11,96

2.52

1,88

1,88

,61

566

27.8ii

12,02

2ol42

1.85

1.85

.57

567

27.

.72

11,96

5o72

ii,90

Iio90

.82

30°

568

27.

.72

11,96

5.67

li,90

ii,90

.77

%9

27.

.72

11,96

7.03

6,30

6,30

.73

570

27<

.72

11,96

6.37

5.63

5.63

.7ii

571

27.

.90

12, OU

li,00

3.30

3.30

.70

63

TABLE VI
Reduction of Experimental Data, Correlating Cj- and X

= -9^ 1^ b2 = .02777 ft.= q = l^V^ SiH^t:

sec.

^L = q b^ X

Pict. V C^ q X Xb^ qXb^ L Cj^

5U9 27o72 11.96 7ii5.35 2.30 .06387 1;7.6055 10.61 .223

5^0 27.72 11.96 7ii5<.35 2.30 .06387 I47.6055 10.61 .223

^51 27.78 11.99 7ii8.58 5.07 .II1O8 105. iiO 18.05 .171

552 27.78 11.99 7U8.58 5.05 .lii02 IOI1.95 18.05 .172

553 27.72 11.96 7li5.35 6.55 0I819 135.58 21.8a .161
55ii 27.72 11.96 7li5.35 6,li3 .1786 133.12 21.81^ .l6ii
9?"^ "^l^l^ 11.99 71^8,58 7.22 .2005 150.09 2ii.71 .165

556 27.78 11,99 7ii8.58 7.1iii .2066 15I4.66 2ii.71 .160

557 27.72 11.96 7^5.35 2.68 .07i;i42 55.ii7 lii.8l .267

558 27.72 11,96 7ii5.35 2.63 ,073014 51^.14; li4.8l .272

559 27.72 11.96 7ii5o35 5.87 .1630 121. I49 27.5ii .227

560 27.72 11,96 7ii5.35 5.82 .1616 120. ii5 27.51; .229

561 27.66 11, 9U 7h2,13 7.25 .2013 lh9.39 32.76 .219

562 27.72 11,96 7i45.35 7.12 .1977 II17.36 32.76 .222

563 27.72 11,96 7li5.35 3o8o ,1055 78.63 19.93 .253
561^ 27,72 11.96 7ii5.35 3. 80 .1055 78,63 19.93 .253

565 27.72 11.96 7li5o35 2,52 .06998 52.16 18„93 .363

566 27. 81^ 12,02 751.82 2oii2 ,06720 50.52 18.93 .375

567 27.72 llo96 7)45.35 5.72 ,1588 118.36 36.69 .310

568 27,72 llo96 7i45.35 5.67 .1575 117.39 36.69 .313

569 27.72 11,96 7ii5.35 7.03 .1952 1145. ii9 l4i4.10 ,303

570 27.72 11.96 7li5.35 6,37 0I769 131.85 liO„28 ,305

571 27.90 12.0I4 755.06 I4.00 ollll 83,88 27.73 .331

64

TABLE VII

Tabulation of Computed and Experimental Results for
the Gravity Independent Lift Coefficient
125 4 56 78

Definit:

ion of

Definition of

ift Based

on V

A Based on c^
e

X r c^

C

L Exp

c

L Comp

L Comp
^L Exp

L Comp

L Comp
L Exp

,30 2

° 21,44

.031

,067

2,16

,072

2.32

.40

18,51

,031

,060

1.935

,065

2.10

.45

24,61

,031

,058

1,87

.063

2,03

.62

20,53

,033

,049

1,485

,052

1.58

.68

20,74

,029

,046

1,585

,049

1.69

.70

24,92

,029

,045

1,55

,048

1,65

,87

14,55

,023

,041

1,78

,041

1,78

.87

22.88

,028

,041

1,46

,041

1,46

1,17

20,04

,027

,035

1.30

,033

1.22

1.57

25,01

,022

,02 9

1,32

,027

1,23

2,54

13,37

,019

,022

1,16

,020

1,05

3,00

13,66

,015

,020

1,33

,018

1,20

3,00

13,72

,015

,020

1,33

,018

1,20

3,00

16,99

,015

,020

1,33

,018

1,20

3,00

21,87

,015

,020

1,33

.018

1,20

4,79

12,20

,012

,015

1,25

,012

1,00

6,50

25,01

,009

,012

1,33

,008

,89

6,50

18.64

.010

,012

1,20

,008

,80

7,79

13,39

,009

,010

1,11

.005

,555

8,12

23,39

,009

,010

1,11

,005

,555

8,27

12.78

,009

,010

1,11

,005

,555

,25 4

° 14,40

,082

,140

1,71

,150

1,83

.35

19.98

,091

,125

1,37

,135

1,48

.38

16,99

,117

,123

1,05

,133

1,14

,38

24,86

,091

,123

1,35

,133

1,45

.50

13,57

,092

,110

1,20

,120

1,31

.54

13,68

,084

,108

1,29

,115

1,37

.62

21,66

,073

,100

1,37

,105

1,44

,85

24,92

,073

,085

1,16

,088

1,20

1,45

21,87

,055

,063

1,14

.057

1,04

1.45

12,81

,054

,063

1,17

,057

1,06

1,50

16,26

,054

,062

1,15

,056

1,04

1,52

16,23

,053

,061

1,15

,055

1,04

1,64

16,37

,048

,059

1,23

,053

1.10

1,66

12,86

,049

,059

1,20

,052

1,06

2,20

24,80

,041

,051

1,24

,042

1,02

3,66

10,97

,029

,039

1,34

,029

1.00

4,07

24,77

,029

,036

1,24

.026

.895

4,54

13,48

,026

,034

1,31

.024

.925

4,63

20,95

,027

,034

1,26

,023

,854

4,66

18,13

,025

,033

1,32

,023

.920

4,82

17.51

,026

,033

1,27

,022

,846

5,14

22,97

,027

,032

1,18

,021

.778

5,54

17,57

,022

,031

1,41

.020

,910

7,16

12,00

,021

,02 9

1,38

,020

,955

8,54

15,46

,019

,026

1.37

.01.2

,630

65,

TABLE VII( Cont'd.)

5 6 7 8
Definition of Definition of

m Based on X

m Based on Of
e

k 1

c

L Comp

L Comp

tr C^

L Exp

L Comp

L Exp

L Comp

c

L Exp

,22 6

° 19.89

.147

,207

1,41

.208

1.42

.25

24,77

,139

.205

1.47

205

1.48

.30

16,68

,153

,194

1,27

.195

1.27

.38

24.92

.163

,181

1.11

.184

1.13

.55

12,69

.143

,158

1.10

.156

1.09

.55

16.32

,145

,158

1.09

,156

1.08

,65

21,90

.123

.150

1.22

,146

1.19

.66

16.27

,122

,149

1.22

,145

1,19

,67

25,10

.131

.148

1.13

.144

1.10

1.32

12,96

.096

,107

1.12

.098

1.02

1.40

24,46

.086

,105

1,22

.094

1,09

1,42

24olO

.088

,104

1.18

.093

1,06

1,42

17.60

,087

,104

1.20

.093

1.07

1.52

20,86

.084

,100

1.19

.090

1.07

1,66

17.37

,076

.095

1,25

,085

1,12

3,00

17.60

,060

.073

1,22

,053

.884

3.05

10.83

.059

,072

1.22

,052

,880

3.22

24,16

,057

,070

1.23

,050

,88

3.32

14,55

.055

,069

1.25

,049

,89

3.35

19.86

,055

.069

1.25

,048

,87

3.41

17.71

.052

.068

1.31

,047

.90

3.79

10.72

.049

.065

1.33

,043

.88

5.20

24.46

,045

.056

1.24

,031

.69

5.57

20,74

,044

.055

1,25

.030

.67

5,62

17.08

.044

,055

1,25

,029

,66

5,62

12,41

,044

,055

1,25

.029

,66

5.75

15.01

.043

.054

1.26

.028

.65

6.29

17.81

.039

.053

1.36

.025

.64

7.29

19.90

,037

,051

1.38

.021

.57

7,66

16.27

.036

.051

1,42

,020

.555

7.79

14.13

,036

,050

1.39

.020

.555

.91 9

° 15.70

,171

.197

1,15

.180

1.05

,91

11.83

.167

,197

1,18

.180

1.08

1.86

18,28

.117

.146

1,25

.117

1.00

3.04

11.80

.091

,121

1,33

.081

.89

3,10

14.10

.090

.120

1,33

.080

.89

3,66

15.40

,084

,114

1,36

.069

,82

5,41

13.78

,070

.104

1.49

.048

,69

5,41

11.98

,071

,104

1.47

.048

,68

7,16

10,82

.066

.096

1,45

.037

,56

7.29

12,30

.066

,096

1,45

.036

.55

.08 12

° 24.98

,256

,455

1.78

,460

1.80

,10

20,04

,318

.445

1.40

,445

1.40

.12

16.96

,371

,435

1.17

.435

1,17

.12

24.95

,285

.435

1.53

.435

1,53

.15

21.59

,305

.425

1.39

.425

1.39

66,

TABLE VII (Cont'd.)

Definit

ion of

Definition of

m Based

L on "V

m Based on 0(
e

X t c^

Ct i.^
L txp

Q

L Comp

Q

L Comp
L Exp

L Comp

L Comp
L Exp

,20 12

° 25,16

,302

„400

1,33

,410

1,36

.20

12,75

.393

,400

1.02

.410

1.04

.25

16.23

.324

,385

1,19

,390

1.20

,28

21,59

,294

,380

1,29

o385

1,31

.30

25.25

.291

,370

1,27

.375

1.29

.31

24,98

,198

,367

1.85

,372

1.88

.40

13.79

„280

,345

1,23

.345

1,23

.40

17,32

,318

,345

1,08

.345

1,08

.45

21.14

.276

,333

1,21

.335

1.21

,48

24,40

,254

,328

1,29

.330

1,30

.53

12,26

,268

,319

1,19

,315

1.18

.54

12.22

,264

,318

1,20

.312

1.18

.70

17,54

,257

,295

1,15

,275

1.07

.75

24,40

,238

,288

1,21

,268

1.12

,75

20,07

,240

,288

1,20

,268

1,11

.77

10.80

,236

,285

1,21

,266

1,13

1.15

24,49

,204

,248

1,22

,214

1,05

1.17

20,89

,209

,247

1,18

,213

1,01

1.20

14,95

,206

,245

1,18

,210

1,02

1,22

12.38

,204

,242

1,19

,208

1,02

1.25

12,50

,195

,240

1,23

,205

1,05

1.80

24,28

,164

,210

1,28

,164

1.00

2.00

23,27

,161

,204

1,27

,152

,94

2.05

14,94

,157

,202

1,29

,150

,96

2.07

21,01

,154

,201

1,31

,149

,97

2,62

16,78

,143

,187

1,31

,125

,87

2,66

12,22

,140

,185

1,32

,125

,89

2.90

21,01

,136

.181

1,33

,116

,85

2.92

11,74

.138

,180

1,31

,115

,83

2,97

20.74

,136

.180

1,32

,114

,84

3,00

16,23

,134

.179

1,34

,113

,84

3.07

16.13

,132

,178

1,35

,112

,85

4,05

16,78

,123

,167

1,36

,088

,72

4.07

10,52

,122

,167

1.37

,087

,71

4,12

18,67

,121

,166

1,37

,086

,71

4.20

14,55

,120

,165

1,37

,085

,71

4.25

12,02

,118

ol65

1,40

,085

.72

.12 18

° 25,10

.50Q

,585

1.15

,580

1,14

,18

12,69

.441

,5 60

1,27

,560

1,27

.18

21,75

,450

,560

1,24

.560

1,24

.18

25.50

,473

,560

1,18

,560

1,18

.20

10,13

,622

,550

.885

,550

,88

,25

21,26

,490

,535

i,09

,525

1,07

.26

15,49

,342

,533

1,56

,523

1,53

.28

10,13

.444

,529

1,19

,515

1,16

,28

17,23

,461

,529

1,14

,515

1,11

67,

TABLE VII (Cont'd,)

7 8

Definition of Definition of

m Based on U

m Based on CV
e

X tr c^

L Exp

L Comp

L Comp
L Exp

C^
L Comp

L Comp
^L Exp

.28 18

° 21.01

.448

.529

1,18

,515

1.15

.28

24.43

,433

.529

1.22

,515

1.19

.35

24.40

.511

.505

,987

.487

.95

.40

17.63

,446

,492

1,10

,470

1.05

.40

10.74

.442

,492

1.11

,470

1.06

.42

10.83

.432

,488

1.13

.465

1.07

.44

19.82

.419

,482

1,15

.460

1.10

,58

15,01

.424

,456

1.08

,426

1.01

.58

24,61

,400

.456

1.14

.425

1.06

.60

20.83

,409

,450

1.10

.420

1.03

.65

12,44

,381

,441

1.16

.416

1.09

.90

23.18

.361

.408

1,13

.368

1.02

.92

10.98

.346

,406

1,17

.364

'1,05

,95

14.91

,343

,403

1,17

.361

1.05

1.02

21.01

.312

.395

1,26

.360

1.12

1.35

16.31

.296

.368

1,26

,310

1.05

1.38

20.98

,287

.366

1,27

.307

1.07

1.47

16.20

.276

.362

1.31

.299

1,08

1.50

15.65

.290

.359

1.24

.294

1.01

1.55

12.99

,276

,356

1.29

,289

1,05

1.87

16.78

,266

,343

1.29

.261

.98

1.88

14.61

.265

,343

1,29

.260

.98

1.98

14.70

,249

.339

1,36

.254

1,02

2.02

18.64

,249

.337

1.35

.250

1.00

2.02

16.80

.247

.337

1.36

.250

1.01

2.02

11,90

.253

.337

1,33

.250

.99

2,30

11.96

.223

,328

1,47

,230

1.03

2,30

11.96

,223

,328

1.47

,230

1,03

5.07

11.99

,171

,284

1,66

.127

.74

5.05

11.99

.172

.284

1,65

.128

.74

6.43

11.96

,164

,275

1.68

.104

,63

6.55

11.96

.161

.2 74

1,70

.102

.63

7,22

11.99

,165

,272

1,65

,094

.57

7.44

11.99

.160

.271

1.69

.091

,57

.22 24

° 21.11

.565

.680

1.20

,686

1,21

.22

17.35

.579

.680

1.18

,686

1.19

,24

13,02

.523

.673

1,29

.675

1.29

.25

io,ao

,501

,671

1.34

,675

1,35

.32

17,45

,568

.645

1,14

,650

1.16

,32

20,04

,563

,645

1,15

.650

1,15

,32

24.40

.558

,645

1.16

,650

1,17

.38

24,34

.473

,635

1,34

.636

1.35

.38

10.80

.480

.635

1,32

,636

1.33

.44

24.61

.527

.618

1,17

.615

1.17

.45

20,86

.544

,617

1.13

,614

1,13

.49

12.47

,503

.610

1,21

,608

1,21

,50

15.01

,492

,605

1,23

.604

1.23

68,

TABLE ViKCont'd J

Definition of

Definit

ion of

A Based on

m Based

L on OC
e

A X

L Comp

Q

L Comp

'y

^L Exp

L Comp

^L Exp

Q

L Comp

"^L Exp

,55 24

° 24,95

.510

.594

1,16

.585

1.15

.55

20.77

.448

,594

1.32

.585

1.30

.55

12.41

,453

,594

1.31

,585

2.31

,65

18.39

.484

.578

1.20

,560

1.16

,70

14,88

,467

.570

1.22

,550

1.08

.70

20,92

,459

.570

1.24

,550

1.20

.70

23.33

,458

.570

1.25

.550

1.20

,94

11.65

.434

.539

1.24

,500

1.15

.95

13.42

,423

.539

1.27

.498

1,18

.98

20.62

,419

.536

1.28

,495

1,18

1.00

16.16

,408

.533

1,31

.487

1,19

1.17

16.59

,437

,517

1.18

,457

1.05

1.22

18.54

,416

,514

1.23

,450

1.08

1.22

11.90

.419

,514

1.23

,450

1.07

1,27

.399

.510

1.28

,443

1.11

2.63

11,96

,2 72

.452

1.66

,308

1.13

2.68

llo96

.267

,451

l.,B9

,306

1.15

3.80

11,96

.253

,430

1.70

.245

.97

5.82

11.96

.229

,412

1.80

,175

.77

5.87

11.96

,227

,412

1.81

,174

.77

7,12

11,96

,222

.406

1.83

,148

,67

7.25

11.94

,219

,405

1.85

,146

.67

.11 30

° 21.72

,739

.826

1.12

,835

1.13

,11

16.41

,719

.826

1.15

,835

1.16

.11

12.78

.711

,826

1.16

,835

1.17

.12

24.98

.740

,822

1.11

.830

1.12'

.16

20.86

,796

,814

1.02

,820

1.03

.16

17.63

.771

,814

1.05

,820

1.06

.18

10.13

,692

.807

1.17

.816

1.18

.27

17.45

.674

.783

1.16

.785

1,16

.27

10.89

,665

,783

1.18

.785

1.18

o31

20,10

,578

,770

1.33

.765

1.32

,40

12.41

.622

.749

1.20

.749

1.20

.40

14.91

.623

.749

1.20

,749

1.20

.54

14,98

.598

,"^18

1.20

.711

1.19

.73

13.24

,566

.686

1.21

,668

1.18

.98

11.99

,514

,658

1.28

.6^2

1.21

.98

14.49

.518

.658

1.27

,622

1.20

2.42

12.02

.375

.583

1.55

.443

1.18

2,52

11.96

.363

.581

1.60

.435

1,20

4.00

12,04

.331

.557

1.68

,330

1,06

5,67

11.96

,313

.544

1.73

,253

.81

5,72

11.96

,310

.543

1.75

.252

,81

6.37

11.96

,305

.540

1.77

.234

.77

7.03

11.96

.303

,537

1,77

.217

.72

69

O 00 Oi ^ CO to
rH O r-H e-l CM CM

CM CO O C^
Cr> r-{

to CO ^ to

in o CO c^ to ^

cnas into cj>5* incj)

to in c- CO CO «o

<
E-i

Pi
O
•H
-P
OS

o
-p

CM

I

-P

Ms

;^h'^

cr>

o

o
-P

s^
o

•H

.-p
o
v>
u
u
o
o

CO

o

•H
Vi
•H

r

O CiS

-p PQ

6

o <u
o o

^^ C

o o

•H
P
•H

•H
<Vh
(D
Q

OS
CD

in
to

CO

p
I

in

CO

CO

<^ ^

1J
a>
•p
o

o
o

>^
o

CD

in

Kj<

to

CM

in

o

o

r— t

CJl

o

o

i-H

1— 1

CM

■-I

■*

o

o

o

O

o

o

O CM O to O CD
CO CO •«*< CO C-- Ti*
O O O O O --H

in

in

o

r-l

CM

o

o

CM CM

-^

CD

t-

iH

O 05

00

t-

o

.— 1

(-)

CM

CM

"*

CO CD

•^

CO

c-

in

CM O

C--

C-

o

o

c

o

O

o

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o

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o

r— 1

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r-i

CM

in

in o to ^ CM o

O <-l rH CM CM ^

o o o o o o

^

CM

en

CD

in

in

^t*

03

CD

CO

CO

CO

^

CO

CO

in

CO

O

cr>

CO

o

o

o

o

o

rH

<-l

CM

rH

CM

CD

o

in in

■<;*<

o

CO

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in

CM

CD i-H

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CO

o

O

f-t

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CM

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CO

CD

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in CM

rH

o

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I-H CM

CM

CO

t> r-i

C^

CO

CO

O

^

c-

CM

in

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00

O

tr~

>*

O

O r-i

rH

CM

CM

■<^

'^

co

CO

en

rH

CD

C--

CO

'^

I-H

O O

O

O

O

o

o

o

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o

•-i

t-i

r-i

CM

CM

CO

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r-^

(T>

CO

CD

rH rH

>;»<

CM

o

o

CO

CM

00

-St*

r-\ r-i

CM

CM

CO

•^

in t>-

c^

o

to

CO

C7)

in

c^

CM

O O

O

o

o

o

o o

o

r-i

M

r-i

r-i

CM

CM

CO

to CD

CO

CM

o o

^ o

O CM

in

to

in

r-i

O CM

r-i r-i

CM

to

^ in

CD CO

CD rH

in

a>

CO

t>

CO in

O O

o

O

o o

O O

O rH

rH

r-i

CM

CM

CO CO

CD CO

CO c^

o

CD

CM rH

•^

C^

CM

^

in

CD

CO t~

r-i r-i

CO CO

in

in

CO O)

r-i

CM

CD

<H

CO

CJ>

05 00

O O

o o

O

O

o o

r-i

r-i

r-i

CM

CM

CM

CO CO

CM CM

in

in

r-i

r-i

r-i r-i

CM CM

o

o

00

CD

CM in

CM CM

•^

'^

C^

t>-

r-i r-i

in in

in

in

in

CO

CO CM

O O

o

o

o

o

r-i r-i

r-i r-i

CM

CM

CO

CO

■^ --e*

CD CM

in

t>

CO

r-i

O

CO

O

CD

CO

c-

>-

a>

CO

r-i

CO CO

tv

CD

rH

o

C-

in

CO

o

in

r-i

00

r-i

r-i

r-i

O O

o

o

r-i

r-i

r-i

r-l

CM

CvJ

CO

CO

•^

^

CO

in

0)
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o
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o

?^

^

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3

O

o

1-^ >-^
o o

°^

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CO

O)

CO

CM

o
to

TABLE IX
Values of the Lift Coefficient for Low Aspect Ratio Airfoils

70.

2TTU

1 ■»■

2 XTT

Jf* UX sin

(Cy found from the momentum
equation with m based on od)

X = 2o86

X = 7.16

^L

^L

0(

°L

^L

10

.Ul6

10

.291

10

.2iih

10

.165

20

.917

20

.699

20

.619

20

MS

30

l.ii82

30

1.193

30

1.087

30

.915

Gl =

2 ^2X ^^

IfoC

4

^ (jj: . 1) -HOC

4.

(C, found from the momentum
equation with m based onoC^

X = 1

oc

^L

10

.381

20

,807

30

1.271

c

OC

L

10

.238

20

.508

30

082I1

X =

2.86

OC

^L

10

0I78

20

o388

30

0637

X = 7.U6

G

^

L

10

.076

10

.171

30

.290

Schematic for Virtuol Mass Definitions

Fig._L

m based on r

m bated on a^

Schematic for Spray Thickness,
Stagnation Point, and Spray Root

Fig

l^aius

Sprou R«ak

Stagnation Rjint

Undistiu'bcd Water 3uriac6

Sireamli'nc

.90
.80
.70
.60

.50
.40
.30

.20

■

.10
.

i 1 1 1 ' ■ 1 1
P.nmniiipH Wnliioc rkf Pi we a fminH

OUIII^UICU VUIUOS Ul ^1 "'■ '^ IvUIIU

from the Quodrotic
C, =B(T + 2/7r)+2-A

-V

2

Lift Coefficient, 0,

'^rB(T + 2/Tr)+2-A|'+2(A-BT)
L o J

\

Fig._2-
1 1 1 1 1 1 1

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from the Cubic
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Lift Coefficient

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Schematic for Heave
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Fig.-§-

Undistorbed
Water Surfnce

Modsl

ActaaL Water Surface

Lift of a Two- Dimensional Flat
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Fig.

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Empirically Corrected Values of
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Fig. 10

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'!-'

BOLLAY

WeiNIG

O EXPERIMENT

a • ANGLE OF ATTACK

Cm ' NORMAL FORCE
" COEFFICIENT

COMPARISON of WEINI6 ond BOLLAY AIRFOIL THEORIES
for the EFFECT of LOW ASPECT RATIO

Flg.J2_

LIFT COEFFICIENT vs. ANGLE OF ATTACK
for FLAT PLATE AIRFOILS

Fig. '3

1.2

-

X = 7. 46

1.0

/
/

.8

/ O

/

/ o

L .6

/ o

.4

/o /

/O /

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.^->^l 1 1 1

t

r t t 1 r

10 20 30 40

Theory with

rh bated on

a

Theory with

rh bosed on
C^e
O Experiment

V J L

A = b» - b

a ~ Angle of

Attack

10 20 30 40

Figures for APPENDIX H
Fig.J4_

.>

1 1

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— -

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Fig. 14 (cont'd)

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30

.n\

Farshiug ^^^^eien"
The 1-^T^ co..iiJ-
flat planing surf.ee.

TtiesiS'
F236

^3031
Farshing

Thif lift coefficient of flat
planing surfaces.

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